J. Math. Phys. 60, 022701 (2019); https://doi.org/10.1063/1.5063514 60, 022701

© 2019 Author(s).

Stochastic fractional differential equationsdriven by Lévy noise under CarathéodoryconditionsCite as: J. Math. Phys. 60, 022701 (2019); https://doi.org/10.1063/1.5063514Submitted: 28 September 2018 . Accepted: 03 February 2019 . Published Online: 26 February 2019

Mahmoud Abouagwa , and Ji Li

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Stochastic fractional differential equations drivenby Lévy noise under Carathéodory conditions

Cite as: J. Math. Phys. 60, 022701 (2019); doi: 10.1063/1.5063514Submitted: 28 September 2018 • Accepted: 3 February 2019 •Published Online: 26 February 2019

Mahmoud Abouagwa1,2,a) and Ji Li2

AFFILIATIONS1Department of Mathematical Statistics, Institute of Statistical Studies and Research, Cairo University, Giza 12613, Egypt2School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074,People’s Republic of China

a)Author to whom correspondence should be addressed:mahmoud.aboagwa@cu.edu.eg

ABSTRACTIn this manuscript, we initiate a study on a class of stochastic fractional differential equations driven by Lévy noise. The existenceand uniqueness theorem of solutions to equations of this class is established under global and local Carathéodory conditions. Ouranalysis makes use of the Carathéodory approximation as well as a stopping time technique. The results obtained here generalizethe main results from Pedjeu and Ladde [Chaos, Solitons Fractals 45, 279–293 (2012)], Xu et al. [Appl. Math. Comput. 263, 398–409(2015)], and Abouagwa et al. [Appl. Math. Comput. 329, 143–153 (2018)]. Finally, an application to the stochastic fractional Burgersdifferential equations is designed to validate the theory obtained.

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I. INTRODUCTIONStochastic differential equations (SDEs) have come to play a vital role in many fields of science and engineering. Indeed,

such SDEs are finding a considerable range of applications in mechanics, control, physics, economics, and other areas.1 The exis-tence and uniqueness theorem of solutions to SDEs driven by Gaussian noise has been studied intensively under Lipschitz andnon-Lipschitz conditions by many scientists. Here, we refer to Arnold,2 Mao,3 Ren and Xia,4 Yamada,5 Yamada and Watanabe,6among others. In 1992, under more generalized conditions with Itô’s Lipschitz and Yamada’s non-Lipschitz conditions as spe-cial cases, Taniguchi7 demonstrated the existence of the unique solutions to Gaussian noise SDEs. Since then, fruitful resultshave been achieved on the existence and uniqueness of solutions under Carathéodory (Taniguchi non-Lipschitz) conditions(cf. Refs. 8–16).

On the other hand, the fractional stochastic differential equations (FSDEs) represent a useful tool to explore the hereditary,memory, and hidden properties of the dynamics of complex systems in engineering, viscoelasticity, signal processing, and otherareas (cf. Ref. 17 and references therein). Different from the integer order equations, FSDEs play a circular role in describingdynamical behavior of many real world phenomena.17 Recently, researchers have shown an increasing interest in studying theexistence and uniqueness of solutions for FSDEs (cf. Refs. 17–20). Pedjeu and Ladde21 established the existence and uniquenessof solutions to a class of Itô-Doob stochastic fractional differential equations (SFDEs) driven by Gaussian noise with uniformLipschitz coefficients by the method of successive approximation. Very recently, motivated by Refs. 4, 5, and 22, Abouagwaet al.23 studied the existence and stability results of the unique solution to non-Lipschitz SFDEs driven by Gaussian noise bymeans of Carathéodory approximation, which has been generalized under Taniguchi non-Lipschitz conditions by Abouagwa andLi.24

However, purely Gaussian noise is no longer appropriate for modeling some real phenomena accompanied with internal orexternal fluctuations with possible jumps.25 Non-Gaussian Lévy noise has come to cover these types of perturbations becauseof its ability in exhibiting long heavy tails of the distribution which make time discontinuity in the sample paths.26 Hence, it

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is significant to focus on SDEs with Lévy noise.27–29 For example, there is a lot of work on Lipschitz SDEs with Lévy noise(see Refs. 25, 26, and 28–31). The existence of the unique solution to jump-diffusion stochastic differential equations with localnon-Lipschitz coefficients is proved by Wang et al.32 Recently, Xu et al.22 established the existence and mean square stability ofthe unique solution to SDEs driven by Lévy noise by successive approximation technique.

In this article, we consider a class of SFDEs driven by Lévy noise of the form

dX(t) = b(t, X(t−))dt + σ(t, X(t−))dB(t) +∫|x|<c

F(t, X(t−), x)N(dt, dx)

+σ1(t, X(t−))(dt)α , t ∈ [0, T], 0 < α < 1

X(0) = X0 ∈ Rd,

(1)

where the mappings b,σ1 : [0, T] × Rd −→ Rd, σ : [0, T] × Rd −→ Rd×m, and F : [0, T] × Rd × Rd −→ Rd. B(t) = (B1(t), . . ., Bm(t)) isan m-dimensional Ft-adapted standard Brownian motion defined on a complete probability space (Ω,F, (Ft, t > 0), P) equippedwith a normal filtration Ft t>0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets of F).N : R+× (Rd− 0) is an independent Ft-adapted Poisson random measure with a compensator Ñ and intensity measure ν, which iscalled Lévy measure such that Ñ(dt, dx)B N(dt, dx) − ν(dx)dt, ∫Rd\0

x2

1+x2ν(dx) < ∞ and the process ∫ t0 ∫ |x|<c F(s, X(s−), x)N(ds, dx) is an

Rd-valued square integrable martingale with the property P(∫ t0 ∫ |x|<c |F(s, X(s−), x) |2ν(dx)ds < ∞) = 1. X0 is the initial value satisfying

E |X0 |2 < ∞, and the constant c ∈ [0, ∞) is the maximum allowable jump size.

Our motivation of this class of equations comes from three aspects. First, the motivation comes from its applications incomplex dynamic processes in sciences and engineering with internal structural and external environmental perturbations andecological and epidemiological dynamics. The increasing interest in this class of equations comes from its generality (i.e., itincludes many special models studied before). Second, due to the Lévy noise perturbation in this class of equations, it is possible tobe applied in financial economics, stochastic filtering, stochastic control, and stochastic resonance in nonlinear signal processing.Finally, the motivation has increased from its applications in modelling real world phenomena where nonlocal property is neededdue to the fractional term.

Overall, it seems that the existence and uniqueness of solutions to SFDEs driven by Lévy noise (1) have scarcely been inves-tigated. Therefore, our aim in this paper is to concern this issue under global and local Carathéodory conditions by using theCarathéodory approximation. It is worth pointing out that comparing with Picard’s successive approximation technique,21,22,25the advantage of using Carathéodory approximation technique is that we do not need to compute X1(t), . . ., Xk−1(t) to computeXk(t). In fact, we can compute Xk(t) directly over intervals of length 1

k (cf. Ref. 3). It is noted that our results are new even whenthe coefficients appeared in Eq. (1) satisfy Lipschitz condition which is a special case of the proposed one. Our novelty is that theresults appeared in Ref. 7 and 21–25 are generalized and improved.

The rest of this manuscript is organized as follows: In Sec. II, some preliminaries and assumptions will be introduced.Section III is devoted to the proof of the existence and uniqueness theorems, followed by an illustrative application in Sec. IV.Finally, the conclusion is given in Sec. V.

Throughout this article, C with or without indexes will denote different constants, whose values may change in differentplaces.

II. PRELIMINARIESIn this section, we mention some preliminaries and state some assumptions needed to establish our results.

Lemma 2.1. (Ref. 33). Let g(t) be a continuous function, then its integration with respect to (dt)α , 0 < α 6 1, is defined by∫ t

0g(s)(ds)α = α

∫ t

0(t − s)α−1g(s)ds.

Definition 2.1. A continuous and Ft adapted Rd-valued stochastic process x(t)06t6T is called a unique solution for Eq. (1) if ithas the following properties:

(i) for arbitrary t ∈ [0, T], Pω : ∫ t0 |x(s) |2ds < ∞ = 1 and almost surely

X(t) = X0 +∫ t

0b(s, X(s−))ds +

∫ t

0σ(s, X(s−))dB(s)

+∫ t

0

∫|x|<c

F(s, X(s−), x)N(ds, dx) + α∫ t

0

σ1(s, X(s−))(t − s)1−α

ds, (2)

(ii) for any other solution x(t), we have Px(t) = x(t), ∀ 0 6 t 6 T = 1.

J. Math. Phys. 60, 022701 (2019); doi: 10.1063/1.5063514 60, 022701-2

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In this article, we will work under the following assumptions:

Hypothesis 1. (The growth condition) There exists a function Γ(t, v) : [0, T] × R+ −→ R+ such that

(1a) Γ(t, v) is locally integrable in t for any fixed v ∈ R+ and is continuous monotone non-decreasing concave in v for every fixedt ∈ [0, T],

(1b) for any fixed t ∈ [0, T] and Y ∈ Rd, the functions b(t, Y), σ(t, Y), F(t, Y, x), and σ1(t, Y) are continuous in Y and satisfy

|b(t, Y) |2 + | |σ(t, Y) | |2 +∫|x|<c

|F(t, Y, x) |2ν(dx) + |σ1(t, Y) |2 6 Γ(t, |Y |2),

(1c) for any positive constant γ, the deterministic differential equation

dvdt= γΓ(t, v), 0 6 t 6 T,

has a global solution for any initial value v0.

Hypothesis 2. (The global condition) There exists a function R(t, v) : [0, T] × R+ −→ R+ such that

(2a) R(t, v) is locally integrable in t for all fixed v ∈ R+ and is continuous non-decreasing concave in v for any fixed t > 0 withthe property R(t, 0) = 0,

(2b) for any fixed t ∈ [0, T] and Y1, Y2 ∈ Rd, the following inequality is satisfied:

|b(t, Y1) − b(t, Y2) |2 + | |σ(t, Y1) − σ(t, Y2) | |2 +∫|x|<c

|F(t, Y1, x) − F(t, Y2, x) |2ν(dx)

+ |σ1(t, Y1) − σ1(t, Y2) |2 6 R(t, |Y1 − Y2 |2),

(2c) if a non-negative continuous function Z(t) satisfies

Z(t) 6 K∫ t

0R(s, Z(s))ds, 0 6 t 6 T,

where K is a positive constant, then Z(t) ≡ 0 for every t ∈ [0, T].

Hypothesis 3. (The local condition) For any integer N > 0, there exists a function RN(t, v) : [0, T] × R+ −→ R+ such that

(3a) RN(t, v) is locally integrable in t for all fixed v ∈ R+ and is continuous non-decreasing concave in v for all fixed t > 0 withthe property RN(t, 0) = 0,

(3b) for all Y1, Y2 ∈ Rd with the property |Y1|, |Y2 | 6 N and t ∈ [0, T], the following is true:

|b(t, Y1) − b(t, Y2) |2 + | |σ(t, Y1) − σ(t, Y2) | |2 +∫|x|<c

|F(t, Y1, x) − F(t, Y2, x) |2ν(dx)

+ |σ1(t, Y1) − σ1(t, Y2) |2 6 RN(t, |Y1 − Y2 |2),

(3c) if a non-negative continuous function Z(t) satisfies

Z(t) 6 K∫ t

0RN(s, Z(s))ds, 0 6 t 6 T,

where K is a positive constant, then Z(t) ≡ 0 for every t ∈ [0, T].

Example 2.1. Let R(t, v) = λ(t)R(v), 0 6 t 6 T, where β(t) : [0,∞) −→ R+ is locally integrable function and R(v) : R+ −→ R+ iscontinuous and non-decreasing concave function with R(0) = 0, R(v) > 0 for every v > 0 such that ∫0+

1R(v)

dv = ∞. Then, the function

R(t, v) fulfills condition 2. Now, we give two concrete examples of the function R. Assume D > 0 and 0 < δ < 1 be sufficiently small.Define

R1(v) = Dv, v > 0,

R2(v) =

v log(v−1), 0 6 v 6 δ,

δ log(δ−1) + ´R2(δ−)(v − δ), v > δ,

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where ´R2 denotes the first derivative of R2. It is well shown that they are all non-decreasing concave functions satisfying ∫0+1

Ri(v)dv =

+∞, i = 1, 2. Particularly, it is shown that the Lipschitz condition is a special case of our proposed condition.

Example 2.2. The following functions meet Hypothesis 2c:

R(t, v) = minξvtδ

,τvθ, ξ ,τ > 0; δ, θ 6 1, ξτ(1 − θ) < 1.

III. THE MAIN RESULTSIn this section, we will establish the existence and uniqueness of the global and local solutions for SFDEs driven by Lévy noise

(1) under Hypotheses 1-3. For this purpose, we define the Carathéodory approximation as follows. For any integer k > 1, defineXk(t) = X0 for all −1 6 t 6 0 and

Xk(t) = X0 +∫ t

0b(s, Xk(s −

1k

))ds +∫ t

0σ(s, Xk(s −

1k

))dB(s)

+∫ t

0

∫|x|<c

F(s, Xk(s −1k

), x)N(ds, dx) + α∫ t

0

σ1(s, Xk(s − 1k ))

(t − s)1−αds, (3)

for t ∈ [0, T].To prove our main results, we need to prove two lemmas. We start by proving the uniform boundedness property for the

sequence of stochastic processes Xk(t) given by Eq. (3).

Lemma 3.1. Assume X0 be F0-measurable, Rd-valued random variable independent of the Wiener process B(t) and the compen-sated Poisson random measure Ñ such that E |X0 |

2 < ∞. Assume Hypothesis 1 hold. Then, for any t ∈ [0, T] and α ∈ ( 12 , 1), the sequence

Xk(t), k > 1 is uniformly bounded.

Proof. By using the simple inequality

|x1 + x2 + · · · + xk |2 6 k( |x1 |

2 + |x2 |2 + · · · + |xk |

2), (4)

and Eq. (3), we have

E*,

sup06s6t

|Xk(s) |2+-6 5E |X0 |

2 + 5E*,

sup06s6t

∫ s

0b(u, Xk(u −

1k

))du

2+-

+5E*,

sup06s6t

∫ s

0σ(u, Xk(u −

1k

))dB(u)

2+-

+5E*,

sup06s6t

∫ s

0

∫|x|<c

F(u, Xk(u −1k

), x)N(du, dx)

2+-

+5α2E*.,

sup06s6t

∫ s

0

σ1(u, Xk(u − 1k ))

(t − u)1−αdu

2+/-

. (α ∈ (1/2, 1)).

By Hölder’s and Doob’s martingale inequalities and Itô-isometry, we obtain

E*,

sup06s6t

|Xk(s) |2+-6 5E |X0 |

2 + 5tE∫ t

0|b(s, Xk(s −

1k

)) |2ds

+20E∫ t

0|σ(s, Xk(s −

1k

)) |2ds

+20E∫ t

0

∫|x|<c

|F(s, Xk(s −1k

), x) |2ν(dx)ds

+5α2t2α−1

2α − 1E∫ t

0|σ1(s, Xk(s −

1k

)) |2ds.

In terms of Hypothesis 1b, we deduce

J. Math. Phys. 60, 022701 (2019); doi: 10.1063/1.5063514 60, 022701-4

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E*,

sup06s6t

|Xk(s) |2+-6 5E |X0 |

2 + 5(8 + T +

α2T2α−1

2α − 1

)E∫ t

0Γ

(s, |Xk(s −

1k

) |2)ds

which, with the help of Jensen inequality, gives

E*,

sup06s6t

|Xk(s) |2+-6 5E |X0 |

2 + 5C∫ t

0Γ*

,s,E*

,sup

06s16s|Xk(s1−) |2+

-+-ds (5)

where C =(8 + T + α2T2α−1

2α−1

).

By Hypothesis 1c and Eq. (5), there exists a solution v(t), t ∈ [0, T], that satisfies

v(t) 6 5E |X0 |2 + 5C

∫ t

0Γ(s, v(s))ds.

By induction, we obtain

E*,

sup06s6t

|Xk(s) |2+-6 v(t) 6 v(T) < ∞, (6)

for all integer k > 1. This proves that the sequence Xk(t), k > 1 is uniformly bounded by a constant, say, C1. Hence, the proof ofLemma 3.1 is completed.

Lemma 3.2. Under Hypothesis 1, for any integer k > 1, α ∈ ( 12 , 1), and 0 6 s < t 6 T, there exists two positive constants C2 and C3

such that

E |Xk(t) − Xk(s) |2 6 C3(t − s) + C2(t − s)2α , (7)

where C2 and C3 depends only on T.

Proof. We begin with

Xk(t) − Xk(s) =∫ t

sb(u, Xk(u −

1k

))du +∫ t

sσ(u, Xk(u −

1k

))dB(u)

+∫ t

s

∫|x|<c

F(u, Xk(u −1k

), x)N(du, dx)

+α∫ s

0

*,

σ1(u, Xk(u − 1k ))

(t − u)1−α−σ1(u, Xk(u − 1

k ))

(s − u)1−α+-du

+α∫ t

s

σ1(u, Xk(u − 1k ))

(t − u)1−αdu.

In view of inequality (4), we get

|Xk(t) − Xk(s) |2 6 4

∫ t

sb(u, Xk(u −

1k

))du

2

+ 4

∫ t

sσ(u, Xk(u −

1k

))dB(u)

2

+4

∫ t

s

∫|x|<c

F(u, Xk(u −1k

), x)N(du, dx)

2

+4α2

∫ s

0

*,

σ1(u, Xk(u − 1k ))

(t − u)1−α−σ1(u, Xk(u − 1

k ))

(s − u)1−α+-du

+∫ t

s

σ1(u, Xk(u − 1k ))

(t − u)1−αdu

2

64∑

i=1

Ii. (8)

First, for I1, I2, and I3, we have by Hölder’s inequality, Itô-isometry, and Hypothesis 1b

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E |I1 | + E |I2 | + E |I3 | 6 4(T − s)E∫ t

s|b(u, Xk(u −

1k

)) |2du

+4E∫ t

s| |σ(u, Xk(u −

1k

)) | |2du

+4E∫ t

s

∫|x|<c

|F(u, Xk(u −1k

), x) |2ν(dx)du

6 4(2 + T − s)∫ t

sΓ*

,u,E*

,sup

06s16u|Xk(s1−) |2+

-+-du.

Given that Γ(t, v) is concave in v for any fixed t > 0 with the property Γ(t, 0) = 0, there exist a(t) > 0, b(t) > 0 such that

Γ(t, v) 6 a(t) + b(t)v, v > 0,∫ T

0a(t)dt < ∞,

∫ T

0b(t)dt < ∞. (9)

By inequality (9) and Lemma 3.1, we get

E |I1 | + E |I2 | + E |I3 | 6 4(2 + T − s)∫ t

s

a(u) + b(u)E*

,sup

06s16u|Xk(s1−) |2+

-

du

6 C3(t − s), (10)

where C3 = (2 + T − s)C2 and C2 = 4[sup06t6T a(t) + C1 sup06t6T b(t)].For I4, we have by Hypothesis 1b, inequality (9), and Lemma 3.1,

E |I4 | = 4α2E

∫ s

0

*,

σ1(u, Xk(u − 1k ))

(t − u)1−α−σ1(u, Xk(u − 1

k ))

(s − u)1−α+-du

+∫ t

s

σ1(u, Xk(u − 1k ))

(t − u)1−αdu

2

6 4α2[ sup06t6T

a(t) + C1 sup06t6T

b(t)]

∫ s

0[(t − u)α−1 − (s − u)α−1]du

+∫ t

s(t − u)α−1du

2

6 C2 |tα − sα |2 6 C2(t − s)2α . (11)

Taking expectation to Eq. (8) and substituting from Eqs. (10) and (11), the required result in Eq. (7) will be obtained. Thiscompletes the proof of Lemma 3.2.

Theorem 3.1. Assume X0 be F0-measurable, Rd-valued random variable independent of the Wiener process B(t) and the com-pensated Poisson random measure Ñ such that E |X0 |

2 < ∞. Assume Hypotheses 1 and 2 hold. Then for any α ∈ ( 12 , 1), the SFDEs driven

by Lévy noise (1) has a unique solution X(t).

Proof. Existence: Given that T > 0 and k→∞, we claim that

E*,

sup06s6T

|X(s) − Xk(s) |2+-−→ 0.

Note that for m > k > 1, it is routine to obtain

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sup06s6t

|Xm(s) − Xk(s) |2

6 4 sup06s6t

∫ s

0[b(u, Xm(u −

1m

)) − b(u, Xk(u −1k

))]du

2

+4 sup06s6t

∫ s

0[σ(u, Xm(u −

1m

)) − σ(u, Xk(u −1k

))]dB(u)

2

+4 sup06s6t

∫ s

0

∫|x|<c

[F(u, Xm(u −1m

), x) − F(u, Xk(u −1k

), x)]N(du, dx)

2

+4α2 sup06s6t

∫ s

0

[σ1(u, Xm(u − 1m )) − σ1(u, Xk(u − 1

k ))]

(s − u)1−αdu

2

64∑

i=1

Ji. (12)

For J1 and J4, we take expectation, use Hölder’s inequality and inequality (4) to give

E |J1 | + E |J4 | 6 4T∫ t

0E |b(s, Xm(s −

1m

)) − b(s, Xk(s −1k

)) |2ds

+4α2t2α−1

2α − 1

∫ t

0E |σ1(s, Xm(s −

1m

)) − σ1(s, Xk(s −1k

)) |2ds

6 8T∫ t

0E |b(s, Xm(s −

1m

)) − b(s, Xk(s −1m

)) |2ds

+8T∫ t

0E |b(s, Xk(s −

1m

)) − b(s, Xk(s −1k

)) |2ds

+8α2T2α−1

2α − 1E |σ1(s, Xm(s −

1m

)) − σ1(s, Xk(s −1m

)) |2ds

+8α2T2α−1

2α − 1

∫ t

0E |σ1(s, Xk(s −

1m

)) − σ1(s, Xk(s −1k

)) |2ds.

By Hypothesis 2b and Lemma 3.2, we obtain

E |J1 | + E |J4 | 6 8(T +

α2T2α−1

2α − 1

) ∫ t

0R(s,E |Xm(s −

1m

) − Xk(s −1m

) |2)ds

+8(T +

α2T2α−1

2α − 1

) ∫ t

0R(s,E |Xk(s −

1m

) − Xk(s −1k

) |2)ds

6 8(T +

α2T2α−1

2α − 1

) ∫ t

0R*

,s,E*

,sup

06s16 s|Xm(s1−) − Xk(s1−) |2+

-+-ds

+8(T +

α2T2α−1

2α − 1

) ∫ t

0R(s, C3(

1k−

1m

) + C2(1k−

1m

)2α)ds. (13)

We proceed next to the estimate of J2 and J3. We have by Doob’s martingale inequality, Itô-isometry andinequality (4)

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E |J2 | + E |J3 | 6 16E∫ t

0| |σ(s, Xm(s −

1m

)) − σ(s, Xk(s −1k

)) | |2ds

+16E∫ t

0

∫|x|<c

|F(s, Xm(s −1m

), x) − F(s, Xk(s −1k

), x) |2ν(dx)ds

6 32E∫ t

0| |σ(s, Xm(s −

1m

)) − σ(s, Xk(s −1m

)) | |2ds

+32E∫ t

0| |σ(s, Xk(s −

1m

)) − σ(s, Xk(s −1k

)) | |2ds

+32E∫ t

0

∫|x|<c

|F(s, Xm(s −1m

), x) − F(s, Xk(s −1m

), x) |2ν(dx)ds

+32E∫ t

0

∫|x|<c

|F(s, Xk(s −1m

), x) − F(s, Xk(s −1k

), x) |2ν(dx)ds.

By Hypothesis 2b and Lemma 3.2, we conclude

E |J2 | + E |J3 | 6 32∫ t

0R(s,E |Xm(s −

1m

) − Xk(s −1m

) |2)ds

+32∫ t

0R(s,E |Xk(s −

1m

) − Xk(s −1k

) |2)ds

6 32∫ t

0R*

,s,E*

,sup

06s16 s|Xm(s1−) − Xk(s1−) |2+

-+-ds

+32∫ t

0R(s, C3(

1k−

1m

) + C2(1k−

1m

)2α)ds. (14)

Taking expectation to Eq. (12) and substituting from Eqs. (13) and (14), we obtain

E*,

sup06s6t

|Xm(s) − Xk(s) |2+-6 C4

∫ t

0R*

,s,E*

,sup

06s16 s|Xm(s1−) − Xk(s1−) |2+

-+-ds

+C4

∫ t

0R(s, C3(

1k−

1m

) + C2(1k−

1m

)2α)ds,

where C4 = 8(4 + T + α2T2α−1

2α−1

)is a positive constant.

Taking limit as m, k −→ ∞, using the fact that R(t, 0) = 0, Lemma 3.1, and Fatous lemma, we obtain

Z(t) 6 C4

∫ t

0R(s, Z(s))ds, (15)

where

Z(t) = lim supm,k−→∞

E*,

sup06s6 t

|Xm(s) − Xk(s) |2+-.

Finallly, through Eq. (15) and Hypothesis 2c, we immediately get Z(t) ≡ 0, i.e.,

E*,

sup06s6t

|Xm(s) − Xk(s) |2+-−→ 0, as m, k −→ ∞. (16)

Indicating that Xk(t), k > 1 is a Cauchy sequence. According to the Borel-Cantelli lemma, there exists a limit, say, X(t) inL2

(Ω, C([0, T],Rd)

)such that Xk(t) → X(t), as k → ∞ and uniformly for all 0 6 t 6 T. Therefore, putting m → ∞ in Eq. (16), we

conclude

limk−→∞

E*,

sup06s6T

|X(s) − Xk(s) |2+-= 0, (17)

as claimed. Moreover, by Lemma 3.1 and as k→∞, we obtain

E*,

sup06s6t

|X(s) |2+-6 ∞, 0 6 t 6 T,

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which means that X(t) is uniformly bounded. Now, we will prove that the limit X(t) is a solution to Eq. (1). Then, for all 0 6 t 6 T,we have

EX(t) − Xk

(t −

1k

)

2

6 2E |X(t) − Xk(t) |2 + 2EXk(t) − Xk

(t −

1k

)

2

,

which, with the help of Lemma 3.2 and Eq. (17) gives

EX(t) − Xk

(t −

1k

)

2

−→ 0 as k −→ ∞.

Therefore, as k→∞ in Eq. (3), we obtain

X(t) = X0 +∫ t

0b(s, X(s−))ds +

∫ t

0σ(s, X(s−))dB(s)

+∫ t

0

∫|x|<c

F(s, X(s−), x)N(ds, dx) + α∫ t

0

σ2(s, X(s−))(t − s)1−α

ds, 0 6 t 6 T.

That is, X(t) is a solution to Eq. (1) This completes the existence proof.Uniqueness: Let X(t), X(t) be two solutions for Eq. (1), then by the same way, we obtain

E*,

sup06s6 t

|X(s) − X(s) |2+-6 C4

∫ t

0R*

,s,E*

,sup

06s16 s|X(s1−) − X(s1−) |2+

-+-ds,

for all 0 6 t 6 T. Applying Hypothesis 2c again to deduce

E*,

sup06s6 t

|X(s) − X(s) |2+-≡ 0,

for all 0 6 t 6 T. Thus, the uniqueness has been proved and the proof of Theorem 3.1 is then complete.

As a generalization of some well known results,21–23,25 we shall present the following corollary:

Corollary 3.1. Under the following conditions(i)

|b(t, y1) − b(t, y2) |2 + | |σ(t, y1) − σ(t, y2) | |2 +∫|x|<c

|F(t, y1, x) − F(t, y2, x) |2ν(dx)

+ |σ1(t, y1) − σ1(t, y2) |2 6 λ(t)κ( |y1 − y2 |2),

(ii) |b(t, 0)|, |σ(t, 0)|, |F(s, 0, x)|, |σ1(t, 0) | ∈ L2loc([0,∞),R+),

where t ∈ [0,∞), y1,y2 ∈ Rd, λ(t) : R+ −→ R+ is locally integrable function and κ(v) : R+ −→ R+ is continuous monotone non-decreasingand concave function with κ(0) = 0 and ∫0+

1κ (v) dv = ∞, there exists a unique solution X(t), t ∈ [0, T] to Eq. (1).

Proof. We note by condition (i) that

|b(t, y) |2 + | |σ(t, y) | |2 +∫|x|<c

|F(t, y, x) |2ν(dx) + |σ1(t, y) |2

6 2[ |b(t, y) − b(t, 0) |2 + | |σ(t, y) − σ(t, 0) | |2 +∫|x|<c

|F(t, y, x) − F(t, 0, x) |2ν(dx)

+ |σ1(t, y) − σ1(t, 0) |2]

+2[ |b(t, 0) |2 + | |σ(t, 0) | |2 +∫|x|<c

|F(t, 0, x) |2ν(dx) + |σ1(t, 0) |2]

6 2λ(t)κ( |y |2) + 2[ |b(t, 0) |2 + | |σ(t, 0) | |2 +∫|x|<c

|F(t, 0, x) |2ν(dx) + |σ1(t, 0) |2].

Given that κ(v) is concave in v ∈ [0, ∞), there exist two positive constants a, b such that κ(v) 6 av + b. Hence,

|b(t, y) |2 + | |σ(t, y) | |2 +∫|x|<c

|F(t, y, x) |2ν(dx) + |σ1(t, y) |2

6 2aλ(t) |y |2 + 2bλ(t) + 2[|b(t, 0) |2 + | |σ(t, 0) | |2 +

∫|x|<c

|F(t, 0, x) |2ν(dx)

+ |σ1(t, 0) |2]6 γ(t) |y |2 + β(t),

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where we have assumed that γ(t) = 2aλ(t) and β(t) = 2bλ(t) + 2[|b(t, 0)|2 + ||σ(t, 0)||2 + ∫ |x |<c|F(t, 0, x)|2ν(dx) + |σ1(t, 0)|2]. Hence, it isshown from condition (ii) that β(t) and γ(t) are locally integrable in t ∈ [0, ∞).

Next, for every t ∈ [0,∞), we set Γ(t, v) = γ(t)v + β(t), v > 0. Then, it is well shown that Hypotheses 1a-1c and 2a-2c hold. Hence,the desirable result will be obtained by applying Theorem 3.1 and the proof of Corollary 3.4 is completed.

Remark 3.1. If σ1 = 0 in Corollary 3.1. Our results can be reduced to some results in Xu et al.22. If F = 0, λ(t) = K, then we obtainsome results in Abouagwa et al.23. In other words, in this special cases, we generalize some results of Refs. 22 and 23

Remark 3.2. If σ1 = 0, κ(v) = v in Corollary 3.1, we obtain some results in the work of Applebaum.25 If F = 0, λ(t) = K, κ(v) = v, weget some results in the work of Pedjeu and Ladde.21 Therefore, some previous results in Refs. 21 and 25 are generalized and improved.

Next, we relax the global Carathéodory condition given by Hypothesis 2 and replace it by the local one in Hypothesis 3 toobtain the existence and uniqueness of the local solutions to Eq. (1) Hence, Theorem 3.2 arrives.

Theorem 3.2. Assume X0 be F0-measurable, Rd-valued random variable independent of the Wiener process B(t) and the com-pensated Poisson random measure Ñ such that E |X0 |

2 < ∞. Assume Hypotheses 1 and 3 hold. Then for any α ∈ ( 12 , 1), the SFDEs driven

by Lévy noise (1) has a unique solution X(t), 0 6 t 6 T.

Proof. Suppose N > 1 be a natural integer and T0 ∈ (0, T). Also, define the sequence of the truncation functions bN(t, X(t)),σN(t, X(t)), FN(t, X(t), x), and σN

1 (t, X(t)) for (t, X) ∈ [0, T0] × Rd as follows:

bN(t, X(t)) =

b(t, X(t)), |X(t) | 6 N,b(t, NX(t)

|X(t)| ), |X(t) | > N,

σN(t, X(t)) =

σ(t, X(t)), |X(t) | 6 N,σ(t, NX(t)

|X(t)| ), |X(t) | > N,

FN(t, X(t), x) =

F(t, X(t), x), |X(t) | 6 N,F(t, NX(t)

|X(t)| , x), |X(t) | > N,

and

σN1 (t, X(t)) =

σ1(t, X(t)), |X(t) | 6 N,σ1(t,

NX(t)|X(t)| ), |X(t) | > N.

Then, the functions bN(t, X(t)), σN(t, X(t)), FN(t, X(t), x), and σN1 (t, X(t)) satisfy Hypothesis 1 and the following inequality:

|bN(t, u) − bN(t, v) |2 + | |σN(t, u) − σN(t, v) | |2 +∫|x|<c

|FN(t, u, x) − FN(t, v, x) |2ν(dx)

+ |σN1 (t, u) − σN

1 (t, v) |2 6 RN(t, |u − v |2), u, v ∈ Rd, t ∈ [0, T0].

Therefore, by Theorem 3.1, the following equation

XN(t) = X0 +∫ t

0bN(s, XN(s−))ds +

∫ t

0σN(s, XN(s−))dB(s)

+∫ t

0

∫|x|<c

FN(s, XN(s−), x)N(ds, dx) + α∫ t

0

σN1 (s, XN(s−))

(t − s)1−αds (18)

has a unique solution XN(t). Moreover, XN+1(t) is the unique solution of the following equation:

XN+1(t) = X0 +∫ t

0bN+1(s, XN+1(s−))ds +

∫ t

0σN+1(s, XN+1(s−))dB(s)

+∫ t

0

∫|x|<c

FN+1(s, XN+1(s−), x)N(ds, dx)

+α∫ t

0

σN+11 (s, XN+1(s−))

(t − s)1−αds. (19)

Define the stopping timesσN B T0 ∧ inft ∈ [0, T] : |XN(t) | > N,

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σN+1 B T0 ∧ inft ∈ [0, T] : |XN+1(t) | > N + 1,

τN B σN ∧ σN+1.

By Eqs. (18) and (19), we have

XN+1(t) − XN(t) =∫ t

0[bN+1(s, XN+1(s−)) − bN(s, XN(s−))]ds

+∫ t

0[σN+1(s, XN+1(s−)) − σN(s, XN(s−))]dB(s)

+∫ t

0

∫|x|<c

[FN+1(s, XN+1(s−), x) − FN(s, XN(s−), x)]N(ds, dx)

+ α∫ t

0

[σN+11 (s, XN+1(s−)) − σN

1 (s, XN(s−))]

(t − s)1−αds.

Taking the expectation and using inequality (4), it deduces that

E*,

sup06s6t∧τN

|XN+1(s) − XN(s) |2+-

6 4E sup06s6t∧τN

∫ s

0[bN+1(u, XN+1(u−)) − bN(u, XN(u−))]du

2

+4E sup06s6t∧τN

∫ s

0[σN+1(u, XN+1(u−)) − σN(u, XN(u−))]dB(u)

2

+4E sup06s6t∧τN

∫ s

0

∫|x|<c

[FN+1(u, XN+1(u−), x) − FN(u, XN(u−), x)]N(du, dx)

2

+4α2E sup06s6t∧τN

∫ s

0

[σN+11 (u, XN+1(u−)) − σN

1 (u, XN(u−))]

(t − u)1−αdu

2

64∑

i=1

Ii. (20)

Using Hölder’s inequality and rearranging the right-hand side terms by the technique of plus and minus, we obtain

I1 + I4 6 4(t ∧ τN)E∫ t∧τN

0|bN+1(s, XN+1(s−)) − bN(s, XN(s−)) |2ds

+4α2(t ∧ τN)2α−1

2α − 1E∫ t∧τN

0|σN+1

1 (s, XN+1(s−)) − σN1 (s, XN(s−)) |2ds

6 8(t ∧ τN)E∫ t∧τN

0|bN+1(s, XN+1(s−)) − bN+1(s, XN(s−)) |2ds

+8(t ∧ τN)E∫ t∧τN

0|bN+1(s, XN(s−)) − bN(s, XN(s−)) |2ds

+8α2(t ∧ τN)2α−1

2α − 1E∫ t∧τN

0|σN+1

1 (s, XN+1(s−)) − σN+11 (s, XN(s−)) |2ds

+8α2(t ∧ τN)2α−1

2α − 1E∫ t∧τN

0|σN+1

1 (s, XN(s−)) − σN1 (s, XN(s−)) |2ds. (21)

The Doob’s martingale inequality and Itô-isometry imply that

J. Math. Phys. 60, 022701 (2019); doi: 10.1063/1.5063514 60, 022701-11

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I2 + I3 6 16E∫ t∧τN

0|σN+1(s, XN+1(s−)) − σN(s, XN(s−)) |2ds

+16E∫ t∧τN

0

∫|x|<c

|FN+1(s, XN+1(s−), x) − FN(s, XN(s−), x) |2ν(dx)ds

6 32E∫ t∧τN

0|σN+1(s, XN+1(s−)) − σN+1(s, XN(s−)) |2ds

+32E∫ t∧τN

0|σN+1(s, XN(s−)) − σN(s, XN(s−)) |2ds

+32E∫ t∧τN

0

∫|x|<c

|FN+1(s, XN+1(s−), x) − FN+1(s, XN(s−), x) |2ν(dx)ds

+32E∫ t∧τN

0

∫|x|<c

|FN+1(s, XN(s−), x) − FN(s, XN(s−), x) |2ν(dx)ds. (22)

For 0 6 s 6 τN, we have

bN+1(s, XN(s−)) = bN(s, XN(s−)) = b(s, XN(s−)),

σN+1(s, XN(s−)) = σN(s, XN(s−)) = σ(s, XN(s−)),

FN+1(s, XN(s−), x) = FN(s, XN(s−), x) = F(s, XN(s−), x),

σN+11 (s, XN(s−)) = σN

1 (s, XN(s−)) = σ1(s, XN(s−)).

(23)

Taking Eqs. (20)–(23) into account, it follows that

E*,

sup06s6t∧τN

|XN+1(s) − XN(s) |2+-

6 8(8 + T +

α2T2α−1

2α − 1

)E∫ t∧τN

0

|bN+1(s, XN+1(s−)) − bN+1(s, XN(s−)) |2

+ |σN+1(s, XN+1(s−)) − σN+1(s, XN(s−)) |2

+∫|x|<c

|FN+1(s, XN+1(s−), x) − FN+1(s, XN(s−), x) |2ν(dx)

+ |σN+11 (s, XN+1(s−)) − σN+1

1 (s, XN(s−)) |2

ds.

Thus, we obtain

E*,

sup06s6t

|XN+1(s ∧ τN) − XN(s ∧ τN) |2+-

6 8CE∫ t

0

|bN+1(s ∧ τN, XN+1(s ∧ τN)) − bN+1(s ∧ τN, XN(s ∧ τN)) |2

+ |σN+1(s ∧ τN, XN+1(s ∧ τN)) − σN+1(s ∧ τN, XN(s ∧ τN)) |2

+∫|x|<c

|FN+1(s ∧ τN, XN+1(s ∧ τN), x) − FN+1(s ∧ τN, XN(s ∧ τN), x) |2ν(dx)

+ |σN+11 (s ∧ τN, XN+1(s ∧ τN)) − σN+1

1 (s ∧ τN, XN(s ∧ τN)) |2

ds.

Then, Hypothesis 3b and Jensen inequality give

E*,

sup06s6t

|XN+1(s ∧ τN) − XN(s ∧ τN) |2+-

6 8CE∫ t

0RN+1

(s ∧ τN, |XN+1(s ∧ τN) − XN(s ∧ τN) |2

)ds

6 8C∫ t

0RN+1*

,s ∧ τN,E*

,sup

06s16s|XN+1(s1 ∧ τN) − XN(s1 ∧ τN) |2+

-+-ds. (24)

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Furthermore, from Eq. (24) and Hypothesis 3c, one can get

E*,

sup06s6t

|XN+1(s ∧ τN) − XN(s ∧ τN) |2+-= 0,

which yields

XN+1(t) = XN(t), (25)

for all t ∈ [0, T0 ∧ τN]. For every ω ∈ Ω, there exists an integer N0 = N0(ω) > 0 such that 0 < T0 6 τN0 . Define X(t) = XN0 (t), t ∈ [0,T], and since XN(t ∧ τN) = X(t ∧ τN) and by (18), it holds that

X(t ∧ τN) = X0 +∫ t∧τN

0bN(s, X(s−))ds +

∫ t∧τN

0σN(s, X(s−))dB(s)

+∫ t∧τN

0

∫|x|<c

FN(s, X(s−), x)N(ds, dx) + α∫ t∧τN

0

σN1 (s, X(s−))

(t − s)1−αds. (26)

Letting N go to infinity and for all t ∈ [0, T], we get

X(t ∧ T) = X0 +∫ t∧T

0b(s, X(s−))ds +

∫ t∧T

0σ(s, X(s−))dB(s)

+∫ t∧T

0

∫|x|<c

F(s, X(s−), x)N(ds, dx) + α∫ t∧T

0

σ1(s, X(s−))(t − s)1−α

ds.

That is,

X(t) = X0 +∫ t

0b(s, X(s−))ds +

∫ t

0σ(s, X(s−))dB(s)

+∫ t

0

∫|x|<c

F(s, X(s−), x)N(ds, dx) + α∫ t

0

σ1(s, X(s−))(t − s)1−α

ds,

indicating that X(t) is a solution for Eq. (1). So far, the existence proof is completed. The uniqueness proof can be obtained bystopping our process X(t). This completes the proof of Theorem 3.2.

Remark 3.3. Observe that Hypothesis 3b is a generalization of the following condition: (1) (the locally Lipschitz condition) for anyinteger N > 0, there exists an LN > 0 such that for any u, v ∈ Rd with |u |, |v | 6 N,

|b(t, u) − b(t, v) |2 + | |σ(t, u) − σ(t, v) | |2 +∫|x|<c

|F(t, u, x) − F(t, v, x) |2ν(dx)

+ |σ1(t, u) − σ1(t, v) |2 6 LN |u − v |2, t ∈ [0, T]. (27)

It should be mentioned that Pedjeu and Ladde21 studied the existence and uniqueness of solutions to SFDEs driven by Brownian noiseunder Lipschitz condition. If we take F = 0 in Ref. 27, then we obtain a unique local solution to SFDEs with Brownian noise whichgeneralize some previous results.21 If σ1 = 0, the condition is studied by Applebaum25. Hence, our results are a generalization of someresults in Refs. 21 and 25.

Remark 3.4. Note that the existence and uniqueness result in Ref. 24 can be inferred from Theorem 3.1 in case of F = 0 in Eq. (1)and Hypotheses 1 and 2. Moreover, when F = 0 in Eq. (1) and Hypotheses 1 and 3, a similar existence result as Theorem 3.2 can beobtained for non-Lipschitz Itô-Doob stochastic fractional differential equations driven by Gaussian noise and this generalizes someresults in Abouagwa and Li.24

Remark 3.5. We note that Theorems 3.1 and 3.2 are proved when α ∈ (1/2, 1). However, we propose to investigate the addressedproblem for α ∈ (1/2, 1) as a future work.

Remark 3.6. It is noted that the theory of impulsive effects are of interest because of their applications in many physical phe-nomena in nature in which states are changed abruptly at certain moments of time. Recently, the theory of stochastic ( fractional)differential equations with impulses has been examined by many researchers (cf. Refs. 34 and 35 and references therein). By adapting

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the same techniques and ideas investigated in this paper, one can prove the existence theorem of the unique solution to the followingimpulsive stochastic fractional differential equations driven by Lévy noise upon making some appropriate conditions:

dX(t) = b(t, X(t−))dt + σ(t, X(t−))dB(t) +∫|x|<c

F(t, X(t−), x)N(dt, dx)

+σ1(t, X(t−))(dt)α , t ∈ [0, T], t , tk, 0 < α < 1,

∆X(tk) = X(t+k) − X(t−k ) = Ik(X(t−k )), k = 1, 2, . . . , m,

X(0) = X0 ∈ Rd,

with Ik : Rd −→ Rd(k = 1, 2, . . . , m) are bounded functions. Furthermore, the fixed times tk satisfies 0 = t0 < t1 < t2 < · · · < tm < T,X(t+

k) and X(t−k ) refers to the right and left limits of X(t) at time tk. ∆X(tk) represents the jump in the state X at time tk, and Ik denotesthe jump size. The functions b, σ, F, and σ1 are defined as before.

IV. AN EXAMPLEIn this section, an example is presented to illustrate the obtained theory. We consider the following stochastic fractional

Burgers differential equations with Dirichlet boundary conditions driven by Lévy noise of the form

∂∂t u(t, ξ) = f1(t, u(t−, ξ)) + f2(t, u(t−, ξ))dL(t) + f3(t, u(t−, ξ))(dt)α , t ∈ [0, T],

u(t, 0) = u(t, 1), t ∈ [0, T],

u(0, ξ) = u0(ξ),

(28)

where 0 6 ξ 6 1, 0 < α < 1, and u0(ξ) ∈ Rd. L(t) is an m-dimensional Lévy motion, and the functions f1, f3 : [0, T] × Rd −→ Rd andf2 : [0, T] × Rd −→ Rd×m are continuous.

In order to rewrite the above system into the abstract form of Eq. (1), let us recall the following Lévy-Itô decomposition in Rd

(cf. Ref. 25):

L(t) = lt + B(t) +∫|x|<c

xN(dt, dx) +∫|x|<c

xN(dt, dx), (29)

where l = E(L(1) − ∫ |x|>c xN(1, dx)) ∈ Rd, B(t) is a Brownian motion independent of the Poisson random measure N(dt, dx) : [0,∞) ×Rd \ 0 with the compensated Poisson random measure Ñ(dt, dx) = N(dt, dx) − ν(dx)dt, and c ∈ [0, ∞) is a constant.

Using the Lévy-Itô decomposition (29), we can rewrite Eq. (28) as

∂

∂tu(t, ξ) = F(t, u(t−, ξ)) + f2(t, u(t−, ξ))dB(t) +

∫|x|<c

f2(t, u(t−, ξ))xN(dt, dx)

+∫|x|>c

f2(t, u(t−, ξ))xN(dt, dx) + f3(t, u(t−, ξ))(dt)α ,

which can be rewritten in the following more general form:

∂

∂tu(t, ξ) = F(t, u(t−, ξ)) + G(t, u(t−, ξ))dB(t) +

∫|x|<c

H(t, u(t−, ξ))N(dt, dx)

+∫|x|>c

H(t, u(t−, ξ))N(dt, dx) + M(t, u(t−, ξ))(dt)α ,

where F = lf2 + f1.Obviously, the term involving large jumps in the above equation is controlled by H and can be neglected through the inter-

lacing technique (cf. Ref. 25) and we focus on studying the following equation driven by continuous noise interspersed with smalljumps:

∂∂t u(t, ξ) = F(t, u(t−, ξ)) + G(t, u(t−, ξ))dB(t) +

∫|x|<c

H(t, u(t−, ξ))N(dt, dx)

+M(t, u(t−, ξ))(dt)α , t ∈ [0, T], 0 6 ξ 6 1, 0 < α < 1,

u(t, 0) = u(t, 1), t ∈ [0, T],

u(0, ξ) = u0(ξ).Let X(t) (ξ) = u(t, ξ) and

b(t, X(t))(ξ) = F(t, u(t, ξ)),

σ(t, X(t))(ξ) = G(t, u(t, ξ)),

F(t, X(t), x)(ξ) = H(t, u(t, ξ), x),

J. Math. Phys. 60, 022701 (2019); doi: 10.1063/1.5063514 60, 022701-14

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σ1(t, X(t))(ξ) = M(t, u(t, ξ)),

Then, the model (28) can be written in the abstract form of Eq. (1). Further, we can impose some suitable conditions on theabove-defined functions to verify the Hypotheses 1-3. Hence, Theorems 3.1 and 3.2 hold for the system (28).

V. CONCLUSIONThrough this work, we have studied the existence of the unique solutions for Itô-Doob stochastic fractional differential equa-

tions driven by Lévy noise under local and global Carathéodory conditions when α ∈ (1/2, 1). The results are obtained by using theCarathéodory approximation as well as a stopping time technique which validated by an application to the stochastic fractionalBurgers differential equations. Some existing results are generalized and improved. Motivated by the random fluctuations withlong-range dependence,36 our future work is not only finding a unique solution to the addressed problem in this paper when α ∈(0, 1/2) but also studying the existence and stability for stochastic fractional differential equations driven by Rosenblatt process.

ACKNOWLEDGMENTSThe authors wish to thank the anonymous referees for their careful reading of the manuscript and for the clarification

comments which led to an improvement of the presentation of the manuscript. This work was supported by NSF of China (GrantNo. 11771161).

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