Numerical Analysis for Stochastic Partial Differential ... · The theory and application of...
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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 128625, 8 pages http://dx.doi.org/10.1155/2013/128625 Research Article Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps Yan Li 1,2 and Junhao Hu 3 1 Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China 2 College of Science, Huazhong Agriculture University, Wuhan 430074, China 3 College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China Correspondence should be addressed to Junhao Hu; [email protected] Received 3 January 2013; Accepted 21 March 2013 Academic Editor: Xuerong Mao Copyright © 2013 Y. Li and J. Hu. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions. 1. Introduction e theory and application of stochastic differential equa- tions have been widely investigated [1–7]. Liu [2] studied the stability of infinite dimensional stochastic differential equations. For the numerical analysis of stochastic partial differential equations, Gy¨ ongy and Krylov [8] discussed the numerical approximations for linear stochastic partial differ- ential equations in whole space. Jentzen et al. [9] studied the numerical simulations of nonlinear parabolic stochastic par- tial differential equations with additive noise. Kloeden et al. [10] gave the error analysis for the pathwise approximation of a general semilinear stochastic evolution equations. By contrast, stochastic partial differential equations with jumps have begun to gain attention [11–15]. R¨ ockner and Zhang [15] considered the existence, uniqueness, and large deviation principles of stochastic evolution equation with jump. In [12], the successive approximation of neutral SPDEs was studied. ere are few papers on the convergence rate of numerical solutions for stochastic partial differential equations with jump, although there are some papers on the convergence rate of numerical solutions for stochastic differential equations with jump in finite dimensions [16, 17]. Being motivated by the papers [16, 17], we will discuss the convergence rate of Euler-Maruyama scheme for a class of stochastic partial delay equations with jump, where the numerical scheme is based on spatial discretization by Galerkin method and time discretization by using a stochastic exponential integrator. In consequence, we generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial delay equations with jump in infinite dimensions. e rest of this paper is arranged as follows. We give some preliminary results of Euler-Maruyama scheme in Section 2. e convergence rate is discussed in Section 3. 2. Preliminary Results roughout this paper, let (Ω, F,{F } ≥0 , P) be a complete probability space with some filtration {F } ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets). Let (, ⟨⋅, ⋅⟩ , ‖⋅‖ ) and (, ⟨⋅, ⋅⟩ , ‖⋅‖ ) be two real separable Hilbert spaces. We denote by (L(, ), ‖ ⋅ ‖) the family of bounded linear operators. Let >0 and ([−, 0], ) denote the family of right-continuous function and leſt-hand limits from [−, 0] to with the norm ‖‖ = sup −≤≤0 ‖()‖ . F 0 ([−, 0], ) denotes the fam- ily of almost surely bounded, F 0 -measurable, ([−, 0], )- valued random variables. For all ≥0, = {( + ) : − ≤ ≤ 0} is regarded as ([−, 0], )-valued stochastic process.
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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 128625 8 pageshttpdxdoiorg1011552013128625
Research ArticleNumerical Analysis for Stochastic Partial Differential DelayEquations with Jumps
Yan Li12 and Junhao Hu3
1 Department of Control Science and Engineering Huazhong University of Science and Technology Wuhan 430074 China2 College of Science Huazhong Agriculture University Wuhan 430074 China3 College of Mathematics and Statistics South-Central University for Nationalities Wuhan 430074 China
Correspondence should be addressed to Junhao Hu hujunhaoyahoocomcn
Received 3 January 2013 Accepted 21 March 2013
Academic Editor Xuerong Mao
Copyright copy 2013 Y Li and J HuThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We investigate the convergence rate of Euler-Maruyamamethod for a class of stochastic partial differential delay equations driven byboth Brownian motion and Poisson point processes We discretize in space by a Galerkin method and in time by using a stochasticexponential integrator We generalize some results of Bao et al (2011) and Jacob et al (2009) in finite dimensions to a class ofstochastic partial differential delay equations with jumps in infinite dimensions
1 Introduction
The theory and application of stochastic differential equa-tions have been widely investigated [1ndash7] Liu [2] studiedthe stability of infinite dimensional stochastic differentialequations For the numerical analysis of stochastic partialdifferential equations Gyongy and Krylov [8] discussed thenumerical approximations for linear stochastic partial differ-ential equations in whole space Jentzen et al [9] studied thenumerical simulations of nonlinear parabolic stochastic par-tial differential equations with additive noise Kloeden et al[10] gave the error analysis for the pathwise approximation ofa general semilinear stochastic evolution equations
By contrast stochastic partial differential equations withjumps have begun to gain attention [11ndash15] Rockner andZhang [15] considered the existence uniqueness and largedeviation principles of stochastic evolution equation withjump In [12] the successive approximation of neutral SPDEswas studied There are few papers on the convergencerate of numerical solutions for stochastic partial differentialequations with jump although there are some papers onthe convergence rate of numerical solutions for stochasticdifferential equations with jump in finite dimensions [16 17]
Being motivated by the papers [16 17] we will discussthe convergence rate of Euler-Maruyama scheme for a classof stochastic partial delay equations with jump where the
numerical scheme is based on spatial discretization byGalerkinmethod and timediscretization by using a stochasticexponential integrator In consequence we generalize someresults of Bao et al (2011) and Jacob et al (2009) in finitedimensions to a class of stochastic partial delay equationswith jump in infinite dimensions The rest of this paper isarranged as follows We give some preliminary results ofEuler-Maruyama scheme in Section 2 The convergence rateis discussed in Section 3
2 Preliminary Results
Throughout this paper let (ΩF F119905119905ge0
P) be a completeprobability space with some filtration F
119905119905ge0
satisfying theusual conditions (ie it is right continuous andF
0contains
allP-null sets) Let (119867 ⟨sdot sdot⟩119867 sdot 119867) and (119870 ⟨sdot sdot⟩
119870 sdot 119870) be
two real separable Hilbert spaces We denote by (L(119870119867)
sdot ) the family of bounded linear operators Let 120591 gt 0 and119863 ([minus120591 0]119867) denote the family of right-continuous functionand left-hand limits 120593 from [minus120591 0] to 119867 with the norm120593119863
= supminus120591le120579le0
120593(120579)119867 119863119887F0([minus120591 0]119867) denotes the fam-
ily of almost surely boundedF0-measurable119863 ([minus120591 0]119867)-
valued random variables For all 119905 ge 0 119883119905= 119883(119905 + 120579) minus120591 le
120579 le 0 is regarded as119863([minus120591 0]119867)-valued stochastic process
2 Abstract and Applied Analysis
Let 119879 be a positive constant For given 120591 ge 0 consider thefollowing stochastic partial differential delay equations withjumps
119889119883 (119905) = [119860119883 (119905) + 119891 (119883 (119905) 119883 (119905 minus 120591))] 119889119905
+ 119892 (119883 (119905) 119883 (119905 minus 120591)) 119889119882 (119905)
+ intZ
ℎ (119883 (119905) 119883 (119905 minus 120591) 119906)119873 (119889119905 119889119906)
(1)
on 119905 isin [0 119879] with initial datum 119883(119905) = 120585(119905) isin
119863119887F0([minus120591 0]119867) minus120591 le 119905 le 0 Here (119860119863(119860)) is a self-adjoint
operator on 119867 119882(119905) 119905 ge 0 is 119870-valued F119905119905ge0
-Wienerprocess defined on the probability space ΩF F
119905119905ge0
P
with covariance operator 119876 We assume that minus119860 and thecovariance operator 119876 of the Wiener process have the sameeigenbasis 119890
119898119898ge1
of119867 that is
minus119860119890119898
= 120582119898119890119898
119876119890119898
= 120572119898119890119898 119898 = 1 2 3
(2)
where 120582119898 119898 isin N are the discrete spectrum of minus119860 and
0 le 1205821
le 1205822sdot sdot sdot le lim
119898rarrinfin120582119898
= infin 120572119898 119898 isin N are the
eigenvalues of 119876 Then119882(119905) is defined by
119882(119905) =
infin
sum119899=1
radic120572119898120573119898(119905) 119890119898 119905 ge 0 (3)
where 120573119898(119905) (119898 = 1 2 3 ) is a sequence of real-valued
standard Brownian motions mutually independent of theprobability space (ΩF F
119905119905ge0
P)According to Da Prato and Zabczyk [1] we define
stochastic integrals with respect to the 119876-Wiener process119882(119905) Let 119870
0= 11987612(119870) be the subspace of 119870 with the
inner product ⟨119906 V⟩1198700
= ⟨119876minus12119906 119876minus12V⟩119870 Obviously 119870
0
is a Hilbert space Denote by L02
= L(1198700 119867) the family
of Hilbert-Schmidt operators from 1198700into 119867 with the norm
Ψ2
L02
= tr((Ψ11987612)(Ψ11987612)lowast
)Let Φ (0infin) rarr L0
2be a predictable F
119905-adapted
process such that
int119905
0
EΦ(119904)L02
119889119904 le infin forall119905 gt 0 (4)
Then the 119867-valued stochastic integral int119905
0Φ(119904)119889119882(119904) is a
continuous square martingale Let 119873(119889119905 119889119906) be the Poissonmeasure which is independent of the119876-Wiener process119882(119905)Denote the compensated or centered Poisson measure as
(119889119905 119889119906) = 119873 (119889119905 119889119906) minus 120588119889119905120587 (119889119906) (5)
where 0 lt 120588 lt infin is known as the jump rate and 120587(sdot) is thejumpdistribution (a probabilitymeasure) LetZ isin B(119870minus0)
be the measurable set Denote byP2([0 119879] timesZ 119867) the spaceof all predictable mappings ℎ [0 119879] times Z rarr 119867 for which
int119879
0
intZ
Eℎ (119905 119906)2
119867119889119905120587 (119889119906) lt infin (6)
Then the119867-valued stochastic integral
int119879
0
intZ
ℎ (119905 119906) (119889119905 119889119906) (7)
is a centred square-integrable martingaleWe recall the definition of the mild solution to (1) as
follows
Definition 1 A stochastic process 119883(119905) 119905 isin [0 119879] is calleda mild solution of (1) if
(i) 119883(119905) is adapted to F119905 119905 ge 0 and has cadlag path on
119905 ge 0 almost surely
(ii) for arbitrary 119905 isin [0 119879] P119908 int119905
0119883(119904)
2
119867119889119904 lt infin =
1 and almost surely
119883 (119905) = 119890119905119860120585 (0) + int
119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
(8)
for any119883(119905) = 120585(119905) isin 119863119887F0([minus120591 0]119867) minus120591 le 119905 le 0
For the existence and uniqueness of the mild solution to(1) (see [11]) we always make the following assumptions
(H1) (119860119863(119860)) is a self-adjoint operator on 119867 such thatminus119860 has discrete spectrum 0 le 120582
1le 1205822
le sdot sdot sdot le
lim119898rarrinfin
120582119898
= infin with corresponding eigenbasis119890119898119898ge1
of 119867 In this case 119860 generates a compact 1198620-
semigroup 119890119905119860 119905 ge 0 such that 119890119905119860 le 119890minus120572119905(H2) The mappings 119891 119867 times 119867 rarr 119867 119892 119867 times
119867 rarr L(119870119867) and ℎ 119867 times 119867 times Z rarr 119867 areBorel measurable and satisfy the following Lipschitzcontinuity condition for some constant 119871
1gt 0 and
arbitrary 119909 119910 1199091 1199101 1199092 1199102isin 119867 and 119906 isin Z
1003817100381710038171003817119891 (1199091 1199101) minus 119891 (119909
2 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101minus 119892 (119909
2 1199102))10038171003817100381710038172
L20
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ (1199091 1199101 119906) minus ℎ (119909
2 1199102 119906)
10038171003817100381710038172
119867
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
(9)
This further implies the linear growth condition thatis
1003817100381710038171003817119891(119909 119910)10038171003817100381710038172
119867+1003817100381710038171003817119892(119909 119910)
10038171003817100381710038172
L02
le 1198710(1 + 119909
2
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867) (10)
where
1198710= 2 (119871
2or1003817100381710038171003817119891(0 0)
10038171003817100381710038172
119867or1003817100381710038171003817119892(0 0)
10038171003817100381710038172
L02
) (11)
Abstract and Applied Analysis 3
(H3) There exists 1198712gt 0 satisfying
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867) (12)
for each 119909 119910 isin 119867 and 119906 isin Z
(H4) For 120585 isin 119863119887F0([minus120591 0]119867) there exists a constant119871
3gt 0
such that
E (1003816100381610038161003816120585 (119904) minus 120585 (119905)
10038161003816100381610038162
) le 1198713|119905 minus 119904|
2 119905 119904 isin [minus120591 0] (13)
We now describe our Euler-Maruyama scheme for theapproximation of (1) For any 119899 ge 1 let 120587
119899 119867 rarr
119867119899= span119890
1 1198902 119890
119899 be the orthogonal projection that is
120587119899119909 = sum
119899
119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 119860
119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892
and ℎ119899= 120587119899ℎ
Consider the following stochastic differential delay equa-tions with jumps on119867
119899
119889119883119899(119905) = [119860
119899119883119899(119905) + 119891
119899(119883119899(119905) 119883
119899(119905 minus 120591))] 119889119905
+ 119892119899(119883119899(119905) 119883
119899(119905 minus 120591)) 119889119882 (119905)
+ intZ
ℎ119899(119883119899(119905) 119883
119899(119905 minus 120591) 119906)119873 (119889119905 119889119906)
119883119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(14)
This spatial approximation (14) is called the Galerkinapproximation of (1) Due to the fact that 120587
119899119860119909 =
120587119899119860(sum119899
119894=1⟨119909 119890119894⟩119867119890119894) = minussum
119899
119894=1120582119894⟨119909 119890119894⟩119867119890119894 119909 isin 119867
119899 it follows
that for 119909 isin 119867119899 119860119899119909 = 119860119909 119890119905119860119899119909 = 119890119905119860119909
By (H2) and (H3) and the property of the projectionoperator we have that
1003817100381710038171003817119860119899119909 minus 11986011989911991010038171003817100381710038172
119867=
1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172
119867le 1205822
119899
1003817100381710038171003817119909 minus 1199101003817100381710038171003817119867
1003817100381710038171003817119891119899 (1199091 1199101) minus 119891119899(1199092 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
=1003817100381710038171003817119891 (1199091 1199101) minus 119891 (119909
2 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899 (1199091 1199101 119906) minus ℎ119899(1199092 1199102 119906)
10038171003817100381710038172
119867
=1003817100381710038171003817ℎ (1199091 1199101 119906) minus ℎ (119909
2 1199102 119906)
10038171003817100381710038172
119867
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899(119909 119910 119906)10038171003817100381710038172
119867=
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867)
(15)
for arbitrary 119909 119910 1199091 1199092 1199101 1199102isin 119867119899and 119906 isin Z Hence (14)
admits a unique solution119883119899(119905) on119867119899
We introduce a time discretization scheme for (14) byusing a stochastic exponential integrator For given119879 ge 0 and120591 gt 0 the time-step size Δ isin (0 1) is defined by Δ = 120591119873
for some sufficiently large integer 119873 gt 120591 For any integer119896 ge 0 the time discretization scheme applied to (14) producesapproximations 119884 119899(119896Δ) asymp 119883119899(119896Δ) by forming
119884119899
((119896 + 1) Δ)
= 119890Δ119860119899 119884
119899
(119896Δ) + 119891119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ
+ 119892119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ119882119896
+intZ
ℎ119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591) 119906) Δ119873119896(119906)
119884119899
(120579) = 120587119899120585 (120579) 120579 isin [minus120591 0]
(16)
where Δ119882119896
= 119882((119896 + 1)Δ) minus 119882(119896Δ) and Δ119873119896(119889119906) =
119873((0 (119896 + 1)Δ] 119889119906) minus 119873((0 119896Δ] 119889119906)The continuous-time version of this scheme associated
with (14) is defined by
119884119899(119905)
= 119890119905119860119899119884119899(0) + int
119905
0
119890(119905minuslfloor119904rfloor)119860119899119891
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minuslfloor119904rfloor)119860119899119892
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minuslfloor119904rfloor)119860119899ℎ
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)
times 119873 (119889119904 119889119906)
119884119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(17)
where lfloor119905rfloor = [119905Δ]Δ with [119905Δ] denotes the integer of 119905ΔFrom (16) and (17) we have 119884119899(119896Δ) = 119884
119899
(119896Δ) forevery 119896 ge 0 That is the discrete-time and continuous-timeschemes coincide at the grid points
3 Convergence Rate
In this section we shall investigate the convergence rate of theEuler-Maruyama method In what follows 119862 gt 0 is a genericconstant whose values may change from line to line
Lemma2 Let (H1)ndash(H4) hold then there is a positive constant119862 gt 0which depends on119879 120585 119871
1 1198712 and 119871
3but is independent
of Δ such that
sup0le119905le119879
(E119883 (119905)2
119867)12
or sup0le119905le119879
(E1003817100381710038171003817119884119899(119905)
10038171003817100381710038172
119867)12
le 119862 (18)
4 Abstract and Applied Analysis
Proof Due to the fact that (E sdot 2
119867)12 is a norm we have
from (8) that
(E119883 (119905)2
119867)12
le (E10038171003817100381710038171003817119890119905119860119899120585 (0)
10038171003817100381710038171003817
2
119867)12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
=
4
sum119894=1
119868119894(119905)
(19)
Recall the property of the operator 119860 (see [18])
10038171003817100381710038171003817(minus119860)120575111989011986011990510038171003817100381710038171003817
le 119862119905minus1205751
10038171003817100381710038171003817(minus119860)1205752 (1 minus 119890
119860119905)10038171003817100381710038171003817le 1198621199051205752 120575
1ge 0 120575
2isin [0 1]
(minus119860)120572+120573
119909 = (minus119860)120572(minus119860)120573119909 119909 isin 119863 ((minus119860)
119903)
(20)
for 120572 120573 isin R where 119903 = max120572 120573 120572 + 120573By (H1) and (H2) together with the Minkowski integral
inequality we derive that
1198682(119905) le int
119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
119867)12
119889119904
le 119862int119905
0
1 + (E119883 (119904)2
119867)12
+(E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862 + 119862int119905
0
(E119883 (119904)2
119867)12
119889119904
(21)
By (H1) (H2) and (H3) and using the Ito isometry we have
1198683(119905) + 119868
4(119905)
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
L02
119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)
+ 120588int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
1198710(1 + E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 1198891199041003817100381710038171003817100381710038171003817
2
119867
)
12
(22)
Using Holder inequality and (H3) for the last term of (22)we have
120588(E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le 119862(Eint119905
0
int119885
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + 119901119862radic2119871
2(int119905
0
E119883 (119904)2
119867119889119904)
12
(23)
Moreover by using the Ito isometry and (H3) we obtain that
(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
Eℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + radic2119871
2(int119905
0
(E119883 (119904)2
119867119889119904)
12
(24)
Substituting (23) and (24) into (22) it follows that
1198683(119905) + 119868
4(119905) le 119862 + 119862E
10038171003817100381710038171205851003817100381710038171003817119867 + 119862(int
119905
0
E119883 (119904)2
119867119889119904)
12
(25)
Hence
(E119883 (119905)2
119867)12
le 119862 + 119862E1003817100381710038171003817120585
1003817100381710038171003817119867 + 119862(int119905
0
E119883 (119904)2
119867119889119904)
12
(26)
Abstract and Applied Analysis 5
Applying the Gronwall inequality we have
sup0le119905le119879
(E119883 (119905)2
119867)12
le 119862 (27)
Using the similar argument the second assertion of (18)follows
Lemma 3 Let (H1)ndash(H4) hold for sufficiently small Δ
sup0le119905le119879
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862Δ12
(28)
where 119862 gt 0 is constant dependent on 119879 120585 1198711 1198712 1198713 and 119871
4
while being independent of Δ
Proof For any 119905 isin [0 119879] we have from (8) that
119883 (119905) minus 119883 (lfloor119905rfloor)
= 119890lfloor119905rfloor119860
(119890(119905minuslfloor119905rfloor)119860
minus 1) 120585 (0)
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ intlfloor119905rfloor
0
intZ
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
lfloor119905rfloor
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
=
7
sum119894=1
119869119894(119905)
(29)
Since (E sdot 2
119867)12 is a norm it follows that
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le
7
sum119894=1
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
(30)
Recalling the fundamental inequality 1 minus 119890minus119910 le 119910 119910 gt 0 weget from (H1) that
10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860
minus 1) 119909100381710038171003817100381710038172
119867
=
1003817100381710038171003817100381710038171003817100381710038171003817
infin
sum119894=1
(119890minus120582119894(119905minuslfloor119905rfloor) minus 1) ⟨119909 119890
119894⟩ 119890119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
119867
le (1 minus 119890minus1205821(119905minuslfloor119905rfloor))
2
1199092
119867
le 1205822
1Δ21199092
119867
(31)
Therefore
(E10038171003817100381710038171198691 (119905)
10038171003817100381710038172
119867)12
= (E10038171003817100381710038171003817119890lfloor119905rfloor119860
119890(119905minuslfloor119905rfloor)119860
minus 1 120585 (0)100381710038171003817100381710038172
119867)12
le 1205821(E
1003817100381710038171003817120585 (0)10038171003817100381710038172
119867)12
Δ
(32)
By (H1) (H2) and the Minkowski integral inequality weobtain that
3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 11003817100381710038171003817100381710038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
times (E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ int119905
lfloor119905rfloor
(E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
(33)
Together with (31) we arrive at3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (1205821Δintlfloor119905rfloor
0
119889119904 + Δ)119862 sup0le119905le119879
(E1003817100381710038171003817119891 (119883 (119905) 119883 (119905 minus 120591))
10038171003817100381710038172
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ
(34)
Following the argument of (22) we derive that7
sum119894=4
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesE1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(intlfloor119905rfloor
0
intZ
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
+(int119905
lfloor119905rfloor
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
E1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(int119905
lfloor119905rfloor
intZ
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(35)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Stochastic AnalysisInternational Journal of
![Page 2: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/2.jpg)
2 Abstract and Applied Analysis
Let 119879 be a positive constant For given 120591 ge 0 consider thefollowing stochastic partial differential delay equations withjumps
119889119883 (119905) = [119860119883 (119905) + 119891 (119883 (119905) 119883 (119905 minus 120591))] 119889119905
+ 119892 (119883 (119905) 119883 (119905 minus 120591)) 119889119882 (119905)
+ intZ
ℎ (119883 (119905) 119883 (119905 minus 120591) 119906)119873 (119889119905 119889119906)
(1)
on 119905 isin [0 119879] with initial datum 119883(119905) = 120585(119905) isin
119863119887F0([minus120591 0]119867) minus120591 le 119905 le 0 Here (119860119863(119860)) is a self-adjoint
operator on 119867 119882(119905) 119905 ge 0 is 119870-valued F119905119905ge0
-Wienerprocess defined on the probability space ΩF F
119905119905ge0
P
with covariance operator 119876 We assume that minus119860 and thecovariance operator 119876 of the Wiener process have the sameeigenbasis 119890
119898119898ge1
of119867 that is
minus119860119890119898
= 120582119898119890119898
119876119890119898
= 120572119898119890119898 119898 = 1 2 3
(2)
where 120582119898 119898 isin N are the discrete spectrum of minus119860 and
0 le 1205821
le 1205822sdot sdot sdot le lim
119898rarrinfin120582119898
= infin 120572119898 119898 isin N are the
eigenvalues of 119876 Then119882(119905) is defined by
119882(119905) =
infin
sum119899=1
radic120572119898120573119898(119905) 119890119898 119905 ge 0 (3)
where 120573119898(119905) (119898 = 1 2 3 ) is a sequence of real-valued
standard Brownian motions mutually independent of theprobability space (ΩF F
119905119905ge0
P)According to Da Prato and Zabczyk [1] we define
stochastic integrals with respect to the 119876-Wiener process119882(119905) Let 119870
0= 11987612(119870) be the subspace of 119870 with the
inner product ⟨119906 V⟩1198700
= ⟨119876minus12119906 119876minus12V⟩119870 Obviously 119870
0
is a Hilbert space Denote by L02
= L(1198700 119867) the family
of Hilbert-Schmidt operators from 1198700into 119867 with the norm
Ψ2
L02
= tr((Ψ11987612)(Ψ11987612)lowast
)Let Φ (0infin) rarr L0
2be a predictable F
119905-adapted
process such that
int119905
0
EΦ(119904)L02
119889119904 le infin forall119905 gt 0 (4)
Then the 119867-valued stochastic integral int119905
0Φ(119904)119889119882(119904) is a
continuous square martingale Let 119873(119889119905 119889119906) be the Poissonmeasure which is independent of the119876-Wiener process119882(119905)Denote the compensated or centered Poisson measure as
(119889119905 119889119906) = 119873 (119889119905 119889119906) minus 120588119889119905120587 (119889119906) (5)
where 0 lt 120588 lt infin is known as the jump rate and 120587(sdot) is thejumpdistribution (a probabilitymeasure) LetZ isin B(119870minus0)
be the measurable set Denote byP2([0 119879] timesZ 119867) the spaceof all predictable mappings ℎ [0 119879] times Z rarr 119867 for which
int119879
0
intZ
Eℎ (119905 119906)2
119867119889119905120587 (119889119906) lt infin (6)
Then the119867-valued stochastic integral
int119879
0
intZ
ℎ (119905 119906) (119889119905 119889119906) (7)
is a centred square-integrable martingaleWe recall the definition of the mild solution to (1) as
follows
Definition 1 A stochastic process 119883(119905) 119905 isin [0 119879] is calleda mild solution of (1) if
(i) 119883(119905) is adapted to F119905 119905 ge 0 and has cadlag path on
119905 ge 0 almost surely
(ii) for arbitrary 119905 isin [0 119879] P119908 int119905
0119883(119904)
2
119867119889119904 lt infin =
1 and almost surely
119883 (119905) = 119890119905119860120585 (0) + int
119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
(8)
for any119883(119905) = 120585(119905) isin 119863119887F0([minus120591 0]119867) minus120591 le 119905 le 0
For the existence and uniqueness of the mild solution to(1) (see [11]) we always make the following assumptions
(H1) (119860119863(119860)) is a self-adjoint operator on 119867 such thatminus119860 has discrete spectrum 0 le 120582
1le 1205822
le sdot sdot sdot le
lim119898rarrinfin
120582119898
= infin with corresponding eigenbasis119890119898119898ge1
of 119867 In this case 119860 generates a compact 1198620-
semigroup 119890119905119860 119905 ge 0 such that 119890119905119860 le 119890minus120572119905(H2) The mappings 119891 119867 times 119867 rarr 119867 119892 119867 times
119867 rarr L(119870119867) and ℎ 119867 times 119867 times Z rarr 119867 areBorel measurable and satisfy the following Lipschitzcontinuity condition for some constant 119871
1gt 0 and
arbitrary 119909 119910 1199091 1199101 1199092 1199102isin 119867 and 119906 isin Z
1003817100381710038171003817119891 (1199091 1199101) minus 119891 (119909
2 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101minus 119892 (119909
2 1199102))10038171003817100381710038172
L20
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ (1199091 1199101 119906) minus ℎ (119909
2 1199102 119906)
10038171003817100381710038172
119867
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
(9)
This further implies the linear growth condition thatis
1003817100381710038171003817119891(119909 119910)10038171003817100381710038172
119867+1003817100381710038171003817119892(119909 119910)
10038171003817100381710038172
L02
le 1198710(1 + 119909
2
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867) (10)
where
1198710= 2 (119871
2or1003817100381710038171003817119891(0 0)
10038171003817100381710038172
119867or1003817100381710038171003817119892(0 0)
10038171003817100381710038172
L02
) (11)
Abstract and Applied Analysis 3
(H3) There exists 1198712gt 0 satisfying
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867) (12)
for each 119909 119910 isin 119867 and 119906 isin Z
(H4) For 120585 isin 119863119887F0([minus120591 0]119867) there exists a constant119871
3gt 0
such that
E (1003816100381610038161003816120585 (119904) minus 120585 (119905)
10038161003816100381610038162
) le 1198713|119905 minus 119904|
2 119905 119904 isin [minus120591 0] (13)
We now describe our Euler-Maruyama scheme for theapproximation of (1) For any 119899 ge 1 let 120587
119899 119867 rarr
119867119899= span119890
1 1198902 119890
119899 be the orthogonal projection that is
120587119899119909 = sum
119899
119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 119860
119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892
and ℎ119899= 120587119899ℎ
Consider the following stochastic differential delay equa-tions with jumps on119867
119899
119889119883119899(119905) = [119860
119899119883119899(119905) + 119891
119899(119883119899(119905) 119883
119899(119905 minus 120591))] 119889119905
+ 119892119899(119883119899(119905) 119883
119899(119905 minus 120591)) 119889119882 (119905)
+ intZ
ℎ119899(119883119899(119905) 119883
119899(119905 minus 120591) 119906)119873 (119889119905 119889119906)
119883119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(14)
This spatial approximation (14) is called the Galerkinapproximation of (1) Due to the fact that 120587
119899119860119909 =
120587119899119860(sum119899
119894=1⟨119909 119890119894⟩119867119890119894) = minussum
119899
119894=1120582119894⟨119909 119890119894⟩119867119890119894 119909 isin 119867
119899 it follows
that for 119909 isin 119867119899 119860119899119909 = 119860119909 119890119905119860119899119909 = 119890119905119860119909
By (H2) and (H3) and the property of the projectionoperator we have that
1003817100381710038171003817119860119899119909 minus 11986011989911991010038171003817100381710038172
119867=
1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172
119867le 1205822
119899
1003817100381710038171003817119909 minus 1199101003817100381710038171003817119867
1003817100381710038171003817119891119899 (1199091 1199101) minus 119891119899(1199092 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
=1003817100381710038171003817119891 (1199091 1199101) minus 119891 (119909
2 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899 (1199091 1199101 119906) minus ℎ119899(1199092 1199102 119906)
10038171003817100381710038172
119867
=1003817100381710038171003817ℎ (1199091 1199101 119906) minus ℎ (119909
2 1199102 119906)
10038171003817100381710038172
119867
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899(119909 119910 119906)10038171003817100381710038172
119867=
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867)
(15)
for arbitrary 119909 119910 1199091 1199092 1199101 1199102isin 119867119899and 119906 isin Z Hence (14)
admits a unique solution119883119899(119905) on119867119899
We introduce a time discretization scheme for (14) byusing a stochastic exponential integrator For given119879 ge 0 and120591 gt 0 the time-step size Δ isin (0 1) is defined by Δ = 120591119873
for some sufficiently large integer 119873 gt 120591 For any integer119896 ge 0 the time discretization scheme applied to (14) producesapproximations 119884 119899(119896Δ) asymp 119883119899(119896Δ) by forming
119884119899
((119896 + 1) Δ)
= 119890Δ119860119899 119884
119899
(119896Δ) + 119891119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ
+ 119892119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ119882119896
+intZ
ℎ119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591) 119906) Δ119873119896(119906)
119884119899
(120579) = 120587119899120585 (120579) 120579 isin [minus120591 0]
(16)
where Δ119882119896
= 119882((119896 + 1)Δ) minus 119882(119896Δ) and Δ119873119896(119889119906) =
119873((0 (119896 + 1)Δ] 119889119906) minus 119873((0 119896Δ] 119889119906)The continuous-time version of this scheme associated
with (14) is defined by
119884119899(119905)
= 119890119905119860119899119884119899(0) + int
119905
0
119890(119905minuslfloor119904rfloor)119860119899119891
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minuslfloor119904rfloor)119860119899119892
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minuslfloor119904rfloor)119860119899ℎ
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)
times 119873 (119889119904 119889119906)
119884119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(17)
where lfloor119905rfloor = [119905Δ]Δ with [119905Δ] denotes the integer of 119905ΔFrom (16) and (17) we have 119884119899(119896Δ) = 119884
119899
(119896Δ) forevery 119896 ge 0 That is the discrete-time and continuous-timeschemes coincide at the grid points
3 Convergence Rate
In this section we shall investigate the convergence rate of theEuler-Maruyama method In what follows 119862 gt 0 is a genericconstant whose values may change from line to line
Lemma2 Let (H1)ndash(H4) hold then there is a positive constant119862 gt 0which depends on119879 120585 119871
1 1198712 and 119871
3but is independent
of Δ such that
sup0le119905le119879
(E119883 (119905)2
119867)12
or sup0le119905le119879
(E1003817100381710038171003817119884119899(119905)
10038171003817100381710038172
119867)12
le 119862 (18)
4 Abstract and Applied Analysis
Proof Due to the fact that (E sdot 2
119867)12 is a norm we have
from (8) that
(E119883 (119905)2
119867)12
le (E10038171003817100381710038171003817119890119905119860119899120585 (0)
10038171003817100381710038171003817
2
119867)12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
=
4
sum119894=1
119868119894(119905)
(19)
Recall the property of the operator 119860 (see [18])
10038171003817100381710038171003817(minus119860)120575111989011986011990510038171003817100381710038171003817
le 119862119905minus1205751
10038171003817100381710038171003817(minus119860)1205752 (1 minus 119890
119860119905)10038171003817100381710038171003817le 1198621199051205752 120575
1ge 0 120575
2isin [0 1]
(minus119860)120572+120573
119909 = (minus119860)120572(minus119860)120573119909 119909 isin 119863 ((minus119860)
119903)
(20)
for 120572 120573 isin R where 119903 = max120572 120573 120572 + 120573By (H1) and (H2) together with the Minkowski integral
inequality we derive that
1198682(119905) le int
119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
119867)12
119889119904
le 119862int119905
0
1 + (E119883 (119904)2
119867)12
+(E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862 + 119862int119905
0
(E119883 (119904)2
119867)12
119889119904
(21)
By (H1) (H2) and (H3) and using the Ito isometry we have
1198683(119905) + 119868
4(119905)
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
L02
119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)
+ 120588int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
1198710(1 + E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 1198891199041003817100381710038171003817100381710038171003817
2
119867
)
12
(22)
Using Holder inequality and (H3) for the last term of (22)we have
120588(E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le 119862(Eint119905
0
int119885
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + 119901119862radic2119871
2(int119905
0
E119883 (119904)2
119867119889119904)
12
(23)
Moreover by using the Ito isometry and (H3) we obtain that
(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
Eℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + radic2119871
2(int119905
0
(E119883 (119904)2
119867119889119904)
12
(24)
Substituting (23) and (24) into (22) it follows that
1198683(119905) + 119868
4(119905) le 119862 + 119862E
10038171003817100381710038171205851003817100381710038171003817119867 + 119862(int
119905
0
E119883 (119904)2
119867119889119904)
12
(25)
Hence
(E119883 (119905)2
119867)12
le 119862 + 119862E1003817100381710038171003817120585
1003817100381710038171003817119867 + 119862(int119905
0
E119883 (119904)2
119867119889119904)
12
(26)
Abstract and Applied Analysis 5
Applying the Gronwall inequality we have
sup0le119905le119879
(E119883 (119905)2
119867)12
le 119862 (27)
Using the similar argument the second assertion of (18)follows
Lemma 3 Let (H1)ndash(H4) hold for sufficiently small Δ
sup0le119905le119879
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862Δ12
(28)
where 119862 gt 0 is constant dependent on 119879 120585 1198711 1198712 1198713 and 119871
4
while being independent of Δ
Proof For any 119905 isin [0 119879] we have from (8) that
119883 (119905) minus 119883 (lfloor119905rfloor)
= 119890lfloor119905rfloor119860
(119890(119905minuslfloor119905rfloor)119860
minus 1) 120585 (0)
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ intlfloor119905rfloor
0
intZ
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
lfloor119905rfloor
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
=
7
sum119894=1
119869119894(119905)
(29)
Since (E sdot 2
119867)12 is a norm it follows that
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le
7
sum119894=1
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
(30)
Recalling the fundamental inequality 1 minus 119890minus119910 le 119910 119910 gt 0 weget from (H1) that
10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860
minus 1) 119909100381710038171003817100381710038172
119867
=
1003817100381710038171003817100381710038171003817100381710038171003817
infin
sum119894=1
(119890minus120582119894(119905minuslfloor119905rfloor) minus 1) ⟨119909 119890
119894⟩ 119890119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
119867
le (1 minus 119890minus1205821(119905minuslfloor119905rfloor))
2
1199092
119867
le 1205822
1Δ21199092
119867
(31)
Therefore
(E10038171003817100381710038171198691 (119905)
10038171003817100381710038172
119867)12
= (E10038171003817100381710038171003817119890lfloor119905rfloor119860
119890(119905minuslfloor119905rfloor)119860
minus 1 120585 (0)100381710038171003817100381710038172
119867)12
le 1205821(E
1003817100381710038171003817120585 (0)10038171003817100381710038172
119867)12
Δ
(32)
By (H1) (H2) and the Minkowski integral inequality weobtain that
3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 11003817100381710038171003817100381710038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
times (E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ int119905
lfloor119905rfloor
(E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
(33)
Together with (31) we arrive at3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (1205821Δintlfloor119905rfloor
0
119889119904 + Δ)119862 sup0le119905le119879
(E1003817100381710038171003817119891 (119883 (119905) 119883 (119905 minus 120591))
10038171003817100381710038172
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ
(34)
Following the argument of (22) we derive that7
sum119894=4
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesE1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(intlfloor119905rfloor
0
intZ
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
+(int119905
lfloor119905rfloor
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
E1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(int119905
lfloor119905rfloor
intZ
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(35)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Stochastic AnalysisInternational Journal of
![Page 3: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/3.jpg)
Abstract and Applied Analysis 3
(H3) There exists 1198712gt 0 satisfying
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867) (12)
for each 119909 119910 isin 119867 and 119906 isin Z
(H4) For 120585 isin 119863119887F0([minus120591 0]119867) there exists a constant119871
3gt 0
such that
E (1003816100381610038161003816120585 (119904) minus 120585 (119905)
10038161003816100381610038162
) le 1198713|119905 minus 119904|
2 119905 119904 isin [minus120591 0] (13)
We now describe our Euler-Maruyama scheme for theapproximation of (1) For any 119899 ge 1 let 120587
119899 119867 rarr
119867119899= span119890
1 1198902 119890
119899 be the orthogonal projection that is
120587119899119909 = sum
119899
119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 119860
119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892
and ℎ119899= 120587119899ℎ
Consider the following stochastic differential delay equa-tions with jumps on119867
119899
119889119883119899(119905) = [119860
119899119883119899(119905) + 119891
119899(119883119899(119905) 119883
119899(119905 minus 120591))] 119889119905
+ 119892119899(119883119899(119905) 119883
119899(119905 minus 120591)) 119889119882 (119905)
+ intZ
ℎ119899(119883119899(119905) 119883
119899(119905 minus 120591) 119906)119873 (119889119905 119889119906)
119883119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(14)
This spatial approximation (14) is called the Galerkinapproximation of (1) Due to the fact that 120587
119899119860119909 =
120587119899119860(sum119899
119894=1⟨119909 119890119894⟩119867119890119894) = minussum
119899
119894=1120582119894⟨119909 119890119894⟩119867119890119894 119909 isin 119867
119899 it follows
that for 119909 isin 119867119899 119860119899119909 = 119860119909 119890119905119860119899119909 = 119890119905119860119909
By (H2) and (H3) and the property of the projectionoperator we have that
1003817100381710038171003817119860119899119909 minus 11986011989911991010038171003817100381710038172
119867=
1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172
119867le 1205822
119899
1003817100381710038171003817119909 minus 1199101003817100381710038171003817119867
1003817100381710038171003817119891119899 (1199091 1199101) minus 119891119899(1199092 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
=1003817100381710038171003817119891 (1199091 1199101) minus 119891 (119909
2 1199102)10038171003817100381710038172
119867
or1003817100381710038171003817119892 (1199091 1199101) minus 119892 (119909
2 1199102)10038171003817100381710038172
L02
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899 (1199091 1199101 119906) minus ℎ119899(1199092 1199102 119906)
10038171003817100381710038172
119867
=1003817100381710038171003817ℎ (1199091 1199101 119906) minus ℎ (119909
2 1199102 119906)
10038171003817100381710038172
119867
le 1198711(10038171003817100381710038171199091 minus 119909
2
10038171003817100381710038172
119867+10038171003817100381710038171199101 minus 119910
2
10038171003817100381710038172
119867)
1003817100381710038171003817ℎ119899(119909 119910 119906)10038171003817100381710038172
119867=
1003817100381710038171003817ℎ(119909 119910 119906)10038171003817100381710038172
119867le 1198712(1199092
119867+1003817100381710038171003817119910
10038171003817100381710038172
119867)
(15)
for arbitrary 119909 119910 1199091 1199092 1199101 1199102isin 119867119899and 119906 isin Z Hence (14)
admits a unique solution119883119899(119905) on119867119899
We introduce a time discretization scheme for (14) byusing a stochastic exponential integrator For given119879 ge 0 and120591 gt 0 the time-step size Δ isin (0 1) is defined by Δ = 120591119873
for some sufficiently large integer 119873 gt 120591 For any integer119896 ge 0 the time discretization scheme applied to (14) producesapproximations 119884 119899(119896Δ) asymp 119883119899(119896Δ) by forming
119884119899
((119896 + 1) Δ)
= 119890Δ119860119899 119884
119899
(119896Δ) + 119891119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ
+ 119892119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591)) Δ119882119896
+intZ
ℎ119899(119884119899
(119896Δ) 119884119899
(119896Δ minus 120591) 119906) Δ119873119896(119906)
119884119899
(120579) = 120587119899120585 (120579) 120579 isin [minus120591 0]
(16)
where Δ119882119896
= 119882((119896 + 1)Δ) minus 119882(119896Δ) and Δ119873119896(119889119906) =
119873((0 (119896 + 1)Δ] 119889119906) minus 119873((0 119896Δ] 119889119906)The continuous-time version of this scheme associated
with (14) is defined by
119884119899(119905)
= 119890119905119860119899119884119899(0) + int
119905
0
119890(119905minuslfloor119904rfloor)119860119899119891
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minuslfloor119904rfloor)119860119899119892
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minuslfloor119904rfloor)119860119899ℎ
119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)
times 119873 (119889119904 119889119906)
119884119899(120579) = 120587
119899120585 (120579) 120579 isin [minus120591 0]
(17)
where lfloor119905rfloor = [119905Δ]Δ with [119905Δ] denotes the integer of 119905ΔFrom (16) and (17) we have 119884119899(119896Δ) = 119884
119899
(119896Δ) forevery 119896 ge 0 That is the discrete-time and continuous-timeschemes coincide at the grid points
3 Convergence Rate
In this section we shall investigate the convergence rate of theEuler-Maruyama method In what follows 119862 gt 0 is a genericconstant whose values may change from line to line
Lemma2 Let (H1)ndash(H4) hold then there is a positive constant119862 gt 0which depends on119879 120585 119871
1 1198712 and 119871
3but is independent
of Δ such that
sup0le119905le119879
(E119883 (119905)2
119867)12
or sup0le119905le119879
(E1003817100381710038171003817119884119899(119905)
10038171003817100381710038172
119867)12
le 119862 (18)
4 Abstract and Applied Analysis
Proof Due to the fact that (E sdot 2
119867)12 is a norm we have
from (8) that
(E119883 (119905)2
119867)12
le (E10038171003817100381710038171003817119890119905119860119899120585 (0)
10038171003817100381710038171003817
2
119867)12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
=
4
sum119894=1
119868119894(119905)
(19)
Recall the property of the operator 119860 (see [18])
10038171003817100381710038171003817(minus119860)120575111989011986011990510038171003817100381710038171003817
le 119862119905minus1205751
10038171003817100381710038171003817(minus119860)1205752 (1 minus 119890
119860119905)10038171003817100381710038171003817le 1198621199051205752 120575
1ge 0 120575
2isin [0 1]
(minus119860)120572+120573
119909 = (minus119860)120572(minus119860)120573119909 119909 isin 119863 ((minus119860)
119903)
(20)
for 120572 120573 isin R where 119903 = max120572 120573 120572 + 120573By (H1) and (H2) together with the Minkowski integral
inequality we derive that
1198682(119905) le int
119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
119867)12
119889119904
le 119862int119905
0
1 + (E119883 (119904)2
119867)12
+(E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862 + 119862int119905
0
(E119883 (119904)2
119867)12
119889119904
(21)
By (H1) (H2) and (H3) and using the Ito isometry we have
1198683(119905) + 119868
4(119905)
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
L02
119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)
+ 120588int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
1198710(1 + E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 1198891199041003817100381710038171003817100381710038171003817
2
119867
)
12
(22)
Using Holder inequality and (H3) for the last term of (22)we have
120588(E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le 119862(Eint119905
0
int119885
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + 119901119862radic2119871
2(int119905
0
E119883 (119904)2
119867119889119904)
12
(23)
Moreover by using the Ito isometry and (H3) we obtain that
(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
Eℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + radic2119871
2(int119905
0
(E119883 (119904)2
119867119889119904)
12
(24)
Substituting (23) and (24) into (22) it follows that
1198683(119905) + 119868
4(119905) le 119862 + 119862E
10038171003817100381710038171205851003817100381710038171003817119867 + 119862(int
119905
0
E119883 (119904)2
119867119889119904)
12
(25)
Hence
(E119883 (119905)2
119867)12
le 119862 + 119862E1003817100381710038171003817120585
1003817100381710038171003817119867 + 119862(int119905
0
E119883 (119904)2
119867119889119904)
12
(26)
Abstract and Applied Analysis 5
Applying the Gronwall inequality we have
sup0le119905le119879
(E119883 (119905)2
119867)12
le 119862 (27)
Using the similar argument the second assertion of (18)follows
Lemma 3 Let (H1)ndash(H4) hold for sufficiently small Δ
sup0le119905le119879
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862Δ12
(28)
where 119862 gt 0 is constant dependent on 119879 120585 1198711 1198712 1198713 and 119871
4
while being independent of Δ
Proof For any 119905 isin [0 119879] we have from (8) that
119883 (119905) minus 119883 (lfloor119905rfloor)
= 119890lfloor119905rfloor119860
(119890(119905minuslfloor119905rfloor)119860
minus 1) 120585 (0)
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ intlfloor119905rfloor
0
intZ
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
lfloor119905rfloor
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
=
7
sum119894=1
119869119894(119905)
(29)
Since (E sdot 2
119867)12 is a norm it follows that
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le
7
sum119894=1
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
(30)
Recalling the fundamental inequality 1 minus 119890minus119910 le 119910 119910 gt 0 weget from (H1) that
10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860
minus 1) 119909100381710038171003817100381710038172
119867
=
1003817100381710038171003817100381710038171003817100381710038171003817
infin
sum119894=1
(119890minus120582119894(119905minuslfloor119905rfloor) minus 1) ⟨119909 119890
119894⟩ 119890119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
119867
le (1 minus 119890minus1205821(119905minuslfloor119905rfloor))
2
1199092
119867
le 1205822
1Δ21199092
119867
(31)
Therefore
(E10038171003817100381710038171198691 (119905)
10038171003817100381710038172
119867)12
= (E10038171003817100381710038171003817119890lfloor119905rfloor119860
119890(119905minuslfloor119905rfloor)119860
minus 1 120585 (0)100381710038171003817100381710038172
119867)12
le 1205821(E
1003817100381710038171003817120585 (0)10038171003817100381710038172
119867)12
Δ
(32)
By (H1) (H2) and the Minkowski integral inequality weobtain that
3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 11003817100381710038171003817100381710038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
times (E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ int119905
lfloor119905rfloor
(E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
(33)
Together with (31) we arrive at3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (1205821Δintlfloor119905rfloor
0
119889119904 + Δ)119862 sup0le119905le119879
(E1003817100381710038171003817119891 (119883 (119905) 119883 (119905 minus 120591))
10038171003817100381710038172
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ
(34)
Following the argument of (22) we derive that7
sum119894=4
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesE1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(intlfloor119905rfloor
0
intZ
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
+(int119905
lfloor119905rfloor
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
E1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(int119905
lfloor119905rfloor
intZ
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(35)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Stochastic AnalysisInternational Journal of
![Page 4: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/4.jpg)
4 Abstract and Applied Analysis
Proof Due to the fact that (E sdot 2
119867)12 is a norm we have
from (8) that
(E119883 (119905)2
119867)12
le (E10038171003817100381710038171003817119890119905119860119899120585 (0)
10038171003817100381710038171003817
2
119867)12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
=
4
sum119894=1
119868119894(119905)
(19)
Recall the property of the operator 119860 (see [18])
10038171003817100381710038171003817(minus119860)120575111989011986011990510038171003817100381710038171003817
le 119862119905minus1205751
10038171003817100381710038171003817(minus119860)1205752 (1 minus 119890
119860119905)10038171003817100381710038171003817le 1198621199051205752 120575
1ge 0 120575
2isin [0 1]
(minus119860)120572+120573
119909 = (minus119860)120572(minus119860)120573119909 119909 isin 119863 ((minus119860)
119903)
(20)
for 120572 120573 isin R where 119903 = max120572 120573 120572 + 120573By (H1) and (H2) together with the Minkowski integral
inequality we derive that
1198682(119905) le int
119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
119867)12
119889119904
le 119862int119905
0
1 + (E119883 (119904)2
119867)12
+(E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862 + 119862int119905
0
(E119883 (119904)2
119867)12
119889119904
(21)
By (H1) (H2) and (H3) and using the Ito isometry we have
1198683(119905) + 119868
4(119905)
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038171003817
2
L02
119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)
+ 120588int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
1198710(1 + E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
+ (E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)1003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 1198891199041003817100381710038171003817100381710038171003817
2
119867
)
12
(22)
Using Holder inequality and (H3) for the last term of (22)we have
120588(E10038171003817100381710038171003817100381710038171003817int119905
0
int119885
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le 119862(Eint119905
0
int119885
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + 119901119862radic2119871
2(int119905
0
E119883 (119904)2
119867119889119904)
12
(23)
Moreover by using the Ito isometry and (H3) we obtain that
(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
Eℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le radic1198712(int119905
0
(E119883 (119904)2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le radic1198712radic120591E
10038171003817100381710038171205851003817100381710038171003817119867 + radic2119871
2(int119905
0
(E119883 (119904)2
119867119889119904)
12
(24)
Substituting (23) and (24) into (22) it follows that
1198683(119905) + 119868
4(119905) le 119862 + 119862E
10038171003817100381710038171205851003817100381710038171003817119867 + 119862(int
119905
0
E119883 (119904)2
119867119889119904)
12
(25)
Hence
(E119883 (119905)2
119867)12
le 119862 + 119862E1003817100381710038171003817120585
1003817100381710038171003817119867 + 119862(int119905
0
E119883 (119904)2
119867119889119904)
12
(26)
Abstract and Applied Analysis 5
Applying the Gronwall inequality we have
sup0le119905le119879
(E119883 (119905)2
119867)12
le 119862 (27)
Using the similar argument the second assertion of (18)follows
Lemma 3 Let (H1)ndash(H4) hold for sufficiently small Δ
sup0le119905le119879
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862Δ12
(28)
where 119862 gt 0 is constant dependent on 119879 120585 1198711 1198712 1198713 and 119871
4
while being independent of Δ
Proof For any 119905 isin [0 119879] we have from (8) that
119883 (119905) minus 119883 (lfloor119905rfloor)
= 119890lfloor119905rfloor119860
(119890(119905minuslfloor119905rfloor)119860
minus 1) 120585 (0)
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ intlfloor119905rfloor
0
intZ
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
lfloor119905rfloor
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
=
7
sum119894=1
119869119894(119905)
(29)
Since (E sdot 2
119867)12 is a norm it follows that
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le
7
sum119894=1
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
(30)
Recalling the fundamental inequality 1 minus 119890minus119910 le 119910 119910 gt 0 weget from (H1) that
10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860
minus 1) 119909100381710038171003817100381710038172
119867
=
1003817100381710038171003817100381710038171003817100381710038171003817
infin
sum119894=1
(119890minus120582119894(119905minuslfloor119905rfloor) minus 1) ⟨119909 119890
119894⟩ 119890119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
119867
le (1 minus 119890minus1205821(119905minuslfloor119905rfloor))
2
1199092
119867
le 1205822
1Δ21199092
119867
(31)
Therefore
(E10038171003817100381710038171198691 (119905)
10038171003817100381710038172
119867)12
= (E10038171003817100381710038171003817119890lfloor119905rfloor119860
119890(119905minuslfloor119905rfloor)119860
minus 1 120585 (0)100381710038171003817100381710038172
119867)12
le 1205821(E
1003817100381710038171003817120585 (0)10038171003817100381710038172
119867)12
Δ
(32)
By (H1) (H2) and the Minkowski integral inequality weobtain that
3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 11003817100381710038171003817100381710038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
times (E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ int119905
lfloor119905rfloor
(E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
(33)
Together with (31) we arrive at3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (1205821Δintlfloor119905rfloor
0
119889119904 + Δ)119862 sup0le119905le119879
(E1003817100381710038171003817119891 (119883 (119905) 119883 (119905 minus 120591))
10038171003817100381710038172
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ
(34)
Following the argument of (22) we derive that7
sum119894=4
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesE1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(intlfloor119905rfloor
0
intZ
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
+(int119905
lfloor119905rfloor
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
E1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(int119905
lfloor119905rfloor
intZ
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(35)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Stochastic AnalysisInternational Journal of
![Page 5: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/5.jpg)
Abstract and Applied Analysis 5
Applying the Gronwall inequality we have
sup0le119905le119879
(E119883 (119905)2
119867)12
le 119862 (27)
Using the similar argument the second assertion of (18)follows
Lemma 3 Let (H1)ndash(H4) hold for sufficiently small Δ
sup0le119905le119879
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862Δ12
(28)
where 119862 gt 0 is constant dependent on 119879 120585 1198711 1198712 1198713 and 119871
4
while being independent of Δ
Proof For any 119905 isin [0 119879] we have from (8) that
119883 (119905) minus 119883 (lfloor119905rfloor)
= 119890lfloor119905rfloor119860
(119890(119905minuslfloor119905rfloor)119860
minus 1) 120585 (0)
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119891 (119883 (119904) 119883 (119904 minus 120591)) 119889119904
+ intlfloor119905rfloor
0
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ intlfloor119905rfloor
0
intZ
(119890(119905minuslfloor119905rfloor)119860
minus 1) 119890(lfloor119905rfloorminus119904)119860
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
lfloor119905rfloor
119890(119905minus119904)119860
119892 (119883 (119904) 119883 (119904 minus 120591)) 119889119882 (119904)
+ int119905
lfloor119905rfloor
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
=
7
sum119894=1
119869119894(119905)
(29)
Since (E sdot 2
119867)12 is a norm it follows that
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le
7
sum119894=1
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
(30)
Recalling the fundamental inequality 1 minus 119890minus119910 le 119910 119910 gt 0 weget from (H1) that
10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860
minus 1) 119909100381710038171003817100381710038172
119867
=
1003817100381710038171003817100381710038171003817100381710038171003817
infin
sum119894=1
(119890minus120582119894(119905minuslfloor119905rfloor) minus 1) ⟨119909 119890
119894⟩ 119890119894
1003817100381710038171003817100381710038171003817100381710038171003817
2
119867
le (1 minus 119890minus1205821(119905minuslfloor119905rfloor))
2
1199092
119867
le 1205822
1Δ21199092
119867
(31)
Therefore
(E10038171003817100381710038171198691 (119905)
10038171003817100381710038172
119867)12
= (E10038171003817100381710038171003817119890lfloor119905rfloor119860
119890(119905minuslfloor119905rfloor)119860
minus 1 120585 (0)100381710038171003817100381710038172
119867)12
le 1205821(E
1003817100381710038171003817120585 (0)10038171003817100381710038172
119867)12
Δ
(32)
By (H1) (H2) and the Minkowski integral inequality weobtain that
3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 11003817100381710038171003817100381710038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
times (E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ int119905
lfloor119905rfloor
(E1003817100381710038171003817119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
119867)12
119889119904
(33)
Together with (31) we arrive at3
sum119894=2
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (1205821Δintlfloor119905rfloor
0
119889119904 + Δ)119862 sup0le119905le119879
(E1003817100381710038171003817119891 (119883 (119905) 119883 (119905 minus 120591))
10038171003817100381710038172
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ
(34)
Following the argument of (22) we derive that7
sum119894=4
(E1003817100381710038171003817119869119894 (119905)
10038171003817100381710038172
119867)12
le (intlfloor119905rfloor
0
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesE1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(intlfloor119905rfloor
0
intZ
10038171003817100381710038171003817119890(119905minuslfloor119905rfloor)119860
minus 110038171003817100381710038171003817
210038171003817100381710038171003817119890(lfloor119905rfloorminus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
+(int119905
lfloor119905rfloor
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
E1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038172
L02
119889119904)
12
+ 119862(int119905
lfloor119905rfloor
intZ
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904minus120591) 119906)2
119867120587 (119889119906) 119889119904)
12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(35)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Stochastic AnalysisInternational Journal of
![Page 6: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/6.jpg)
6 Abstract and Applied Analysis
Substituting (32) (34) and (35) into (30) we arrive at
(E119883 (119905) minus 119883 (lfloor119905rfloor)2
119867)12
le 119862(1 + sup0le119905le119879
(E119883 (119905)2
119867)12
)Δ12
(36)
Therefore by Lemma 2 the required assertion (28) follows
Now we state our main result in this paper as follows
Theorem 4 Let (H1)ndash(H4) hold and
radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
) lt 1 (37)
Then
sup0letle119879
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le 119862 120582minus12
119899+ Δ12
(38)
where 119862 gt 0 is a constant dependent on 119879 120585 1198711 1198712 1198713 and
1198714 while being independent of 119899 and Δ
Proof By (8) and (17) we obtain
119883 (119905) minus 119884119899(119905)
= 119890119905119860
(1 minus 120587119899) 120585 (0)
+ int119905
0
119890(119905minus119904)119860
(119891 (119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (119904) 119883 (119904 minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (119904) 119883 (119904 minus 120591))
minus119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119891119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119904
+ int119905
0
119890(119905minus119904)119860
(119892119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591))
minus119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(119892 (119883 (119904) 119883 (119904 minus 120591))
minus119892119899(119883 (119904) 119883 (119904 minus 120591))) 119889119882 (119904)
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) 119891119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119904
+ int119905
0
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)
times 119892119899(119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591)) 119889119882 (119904)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (119904) 119883 (119904 minus 120591) 119906)
minusℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloorminus120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
intZ
119890(119905minus119904)119860
ℎ119899(119883 (lfloor119904rfloor) 119883 (lfloor119904rfloor minus 120591) 119906) minus ℎ
119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
+ int119905
0
int119885
119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
) ℎ119899
times (119884119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591) 119906)119873 (119889119904 119889119906)
=
13
sum119894=1
119870119894(119905)
(39)
Noting that (E sdot 2
119867)12 is a norm we have
(E1003817100381710038171003817119883 (119905) minus 119884
119899(119905)
10038171003817100381710038172
119867)12
le
13
sum119894=1
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
(40)
By (H1) and the nondecreasing spectrum 120582119898119898ge1
it easilyfollows that
E10038171003817100381710038171003817119890119905119860
(1 minus 120587119899) 120585 (0)
10038171003817100381710038171003817119867
= E(infin
sum119898=119899+1
119890minus2120582119898119905⟨120585 (0) 119890
119898⟩2
119867)
12
= E(infin
sum119898=119899+1
119890minus2120582119898119905
1205822119898
1205822
119898⟨120585 (0) 119890
119898⟩2
119867)
12
le1
120582119899
E1003817100381710038171003817119860120585 (0)
1003817100381710038171003817119867
(41)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 7: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/7.jpg)
Abstract and Applied Analysis 7
By (H2) theMinkowski integral inequality and Lemma 2 wehave
(E10038171003817100381710038171198702 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119891 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
119867)12
119889119904
= int119905
0
(Einfin
sum119898=119899+1
119890minus2120582119898(119905minus119904)⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890
119898⟩2
119867)
12
119889119904
le int119905
0
119890minus120582119899(119905minus119904)(E
infin
sum119898=119899+1
⟨119891 (119883 (119904) 119883 (119904 minus 120591)) 119890119898⟩2
119867)
12
119889119904
le 119862int119905
0
119890minus120582119899(119905minus119904)
times 1 + (E119883 (119904)2
119867)12
+ (E119883 (119904 minus 120591)2
119867)12
119889119904
le 119862120582minus1
119899
(42)
Applying (H1) (H2) and Lemma 3 and combining theMinkowski integral inequality and the Ito isometry yield6
sum119894=3
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+119883(119904 minus 120591) minus 119883 (lfloor119904rfloor minus 120591)2
119867))12
119889119904
+ radic1198711int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
(E (1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883 (lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867))12
119889119904
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E (119883 (119904) minus 119883 (lfloor119904rfloor)2
119867
+1003817100381710038171003817119883 (119904minus120591) minus 119884
119899(lfloor119904rfloorminus120591)
10038171003817100381710038172
119867)) 119889119904)
12
+ radic1198711(int119905
0
10038171003817100381710038171003817119890(119905minus119904)11986010038171003817100381710038171003817
2
(E1003817100381710038171003817119883 (lfloor119904rfloor) minus 119884
119899(lfloor119904rfloor)
10038171003817100381710038172
119867
+1003817100381710038171003817119883(lfloor119904rfloor minus 120591) minus 119884
119899(lfloor119904rfloor minus 120591)
10038171003817100381710038172
119867) 119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times int119905
0
119890minus120572(119905minus119904)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905
0
119890minus2120572(119905minus119904)
119889119904)
12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
int119905minus120591
minus120591
119890minus120572(119905minus119904minus120591)
119889119904
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(int119905minus120591
minus120591
119890minus2120572(119905minus119904minus120591)
119889119904)
12
le 119862Δ12
+ radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
times (2120572minus1
+ 2(2120572)minus12
)
(43)
By the Ito isometry and a similar argument to that of (42) wededuce that
(E10038171003817100381710038171198707 (119905)
10038171003817100381710038172
119867)12
le (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899) 119892 (119883 (119904) 119883 (119904 minus 120591))
10038171003817100381710038171003817
2
L02
119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904)E
1003817100381710038171003817119892 (119883 (119904) 119883 (119904 minus 120591))10038171003817100381710038172
L02
119889119904)
12
le 119862120582minus12
119899
(44)
Moreover by (31) (H2) and Lemma 2 and combining theMinkowski integral inequality and the Ito isometry we have
9
sum119894=8
(E1003817100381710038171003817119870119894 (119905)
10038171003817100381710038172
119867)12
le int119905
0
(E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119891119899(119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ (int119905
0
E10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 119890(119904minuslfloor119904rfloor)119860
)10038171003817100381710038171003817
2
times1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867119889119904)
12
le 119862Δint119905
0
(E1003817100381710038171003817119891119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
119867)12
119889119904
+ 119862Δ(int119905
0
E1003817100381710038171003817119892119899 (119884
119899(lfloor119904rfloor) 119884
119899(lfloor119904rfloor minus 120591))
10038171003817100381710038172
L02
119889119904)
12
le 119862Δ
(45)
By (31) and the Ito isometry we obtain that
(E100381710038171003817100381711987010 (119905)
10038171003817100381710038172
119867)12
le (E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) (119889119904 119889119906)10038171003817100381710038171003817100381710038171003817
2
119867
)
12
+ 120588(E10038171003817100381710038171003817100381710038171003817int119905
0
intZ
119890(119905minus119904)119860
(1 minus 120587119899) ℎ
times (119883 (119904) 119883 (119904 minus 120591) 119906) 120587 (119889119906) 11988911990410038171003817100381710038171003817100381710038171003817
2
119867
)
12
le (int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
E
times ℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 8: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/8.jpg)
8 Abstract and Applied Analysis
+ 120588(int119905
0
intZ
10038171003817100381710038171003817119890(119905minus119904)119860
(1 minus 120587119899)10038171003817100381710038171003817
2
timesEℎ (119883 (119904) 119883 (119904 minus 120591) 119906)2
119867120587(119889119906)119889119904)
12
le 119862(int119905
0
119890minus2120582119899(119905minus119904) (E119883 (119904)
2
119867+ E119883 (119904 minus 120591)
2
119867) 119889119904)
12
le 119862120582minus12
119899
(46)
Carrying out the similar arguments to those of (43) and (45)we derive that
(E100381710038171003817100381711987011 (119905)
10038171003817100381710038172
119867)12
+ (E100381710038171003817100381711987012 (119905)
10038171003817100381710038172
119867)12
le 119862Δ12
+ (2120572)minus12
(120588 + 1)
times radic1198711sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(E100381710038171003817100381711987013 (119905)
10038171003817100381710038172
119867)12
le 119862Δ
(47)
As a result putting (41)ndash(47) into (40) gives that
sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
le 119862120582minus12
119899+ 119862Δ12
+ radic1198711(2120572minus1
+ (120588 + 3) (2120572)minus12
)
times sup0le119904le119905
(E1003817100381710038171003817119883 (119904) minus 119884
119899(119904)
10038171003817100381710038172
119867)12
(48)
and therefore the desired assertion follows
Remark 5 For finite-dimensional Euler-Maruyama methodthe condition (37) can be deleted by the Gronwall inequality[16 17]
Acknowledgment
This work is partially supported by National Natural ScienceFoundation of China Under Grants 60904005 and 11271146
References
[1] G Da Prato and J Zabczyk Stochastic Equations in InfiniteDimensions vol 44 of Encyclopedia of Mathematics and ItsApplications Cambridge University Press Cambridge UK1992
[2] K Liu Stability of Infinite Dimensional Stochastic DifferentialEquations with Applications Chapman amp HallCRC BocaRaton Fla USA 2004
[3] Q Luo F Deng J Bao B Zhao and Y Fu ldquoStabilization ofstochastic Hopfield neural network with distributed parame-tersrdquo Science in China F vol 47 no 6 pp 752ndash762 2004
[4] Q Luo F Deng X Mao J Bao and Y Zhang ldquoTheory andapplication of stability for stochastic reaction diffusion systemsrdquoScience in China F vol 51 no 2 pp 158ndash170 2008
[5] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing Chichester UK 2007
[6] Y Shen and JWang ldquoAn improved algebraic criterion for globalexponential stability of recurrent neural networks with time-varying delaysrdquo IEEE Transactions on Neural Networks vol 19no 3 pp 528ndash531 2008
[7] Y Shen and J Wang ldquoAlmost sure exponential stability ofrecurrent neural networks with markovian switchingrdquo IEEETransactions on Neural Networks vol 20 no 5 pp 840ndash8552009
[8] I Gyongy and N Krylov ldquoAccelerated finite difference schemesfor linear stochastic partial differential equations in the wholespacerdquo SIAM Journal on Mathematical Analysis vol 42 no 5pp 2275ndash2296 2010
[9] A Jentzen P E Kloeden and G Winkel ldquoEfficient simulationof nonlinear parabolic SPDEs with additive noiserdquo The Annalsof Applied Probability vol 21 no 3 pp 908ndash950 2011
[10] P E Kloeden G J Lord A Neuenkirch and T Shardlow ldquoTheexponential integrator scheme for stochastic partial differentialequations pathwise error boundsrdquo Journal of Computationaland Applied Mathematics vol 235 no 5 pp 1245ndash1260 2011
[11] J Bao A Truman andC Yuan ldquoStability in distribution ofmildsolutions to stochastic partial differential delay equations withjumpsrdquo Proceedings of The Royal Society of London A vol 465no 2107 pp 2111ndash2134 2009
[12] B Boufoussi and S Hajji ldquoSuccessive approximation of neutralfunctional stochastic differential equations with jumpsrdquo Statis-tics and Probability Letters vol 80 no 5-6 pp 324ndash332 2010
[13] E Hausenblas ldquoFinite element approximation of stochastic par-tial differential equations driven by Poisson random measuresof jump typerdquo SIAM Journal on Numerical Analysis vol 46 no1 pp 437ndash471 200708
[14] S Peszat and J Zabczyk Stochastic Partial Differential Equationswith Levy Noise An Evolution Equation Approach vol 113 ofEncyclopedia of Mathematics and Its Applications CambridgeUniversity Press Cambridge UK 2007
[15] M Rockner and T Zhang ldquoStochastic evolution equations ofjump type existence uniqueness and large deviation princi-plesrdquo Potential Analysis vol 26 no 3 pp 255ndash279 2007
[16] J Bao B Bottcher X Mao and C Yuan ldquoConvergencerate of numerical solutions to SFDEs with jumpsrdquo Journal ofComputational and Applied Mathematics vol 236 no 2 pp119ndash131 2011
[17] N Jacob Y Wang and C Yuan ldquoNumerical solutions ofstochastic differential delay equations with jumpsrdquo StochasticAnalysis and Applications vol 27 no 4 pp 825ndash853 2009
[18] A Pazy Semigroups of Linear Operators and Applications toPartial Differential Equations vol 44 of Applied MathematicalSciences Springer New York NY USA 1983
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 9: Numerical Analysis for Stochastic Partial Differential ... · The theory and application of stochastic differential equa-tions have been widely investigated [1–7]. Liu [2]studied](https://reader033.fdocuments.in/reader033/viewer/2022042908/5f399a78592a3a39b829387e/html5/thumbnails/9.jpg)
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
