Research Article Weak and Strong Limit Theorems for...
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Research Article Weak and Strong Limit Theorems for Stochastic Processes under Nonadditive Probability Xiaoyan Chen 1 and Zengjing Chen 2 1 Graduate Department of Financial Engineering, Ajou University, Suwon 443-749, Republic of Korea 2 School of Mathematics, Shandong University, Jinan 250100, China Correspondence should be addressed to Xiaoyan Chen; cxy [email protected] Received 9 December 2013; Accepted 23 December 2013; Published 13 February 2014 Academic Editor: Litan Yan Copyright © 2014 X. Chen and Z. Chen. 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. is paper extends laws of large numbers under upper probability to sequences of stochastic processes generated by linear interpolation. is extension characterizes the relation between sequences of stochastic processes and subsets of continuous function space in the framework of upper probability. Limit results for sequences of functional random variables and some useful inequalities are also obtained as applications. 1. Introduction Laws of large numbers are the cornerstones of theory of probability and statistics. As we know, under appropriate assumptions, the well-known strong law of large numbers (SLLN for short) states that for a sequence of random vari- ables { } ∞ =1 , its sample mean / := ∑ =1 / converges to a unique constant almost surely in the framework of probability. But many empirical analyses and theoretical works show us that nonadditive probability and nonlinear expectation are very probably faced in economics, finance, number theory, statistics, and many other fields, such as capacity, Choquet integral (see Choquet [1]), (nontrivial) -probability, (nontrivial) -expectation (see El Karoui et al. [2]), and -expectation (see Peng [3]). And for each nonadditive probability, say , we can define many different expectations related to , denoted by . For nonlinear , random variables may have mean uncertainty; that is, [ ] ̸ =− [− ], or variance uncertainty; that is, [ 2 ] ̸ =− [− 2 ]. In such cases, there are many scholars that investigate the limit theorems under or , such as the laws of large numbers, laws of iterated logarithm, central limit theorems under either or , and other related problems. One can refer to Peng [4, 5], Chen and Hu [6], Wu and Chen [7], the papers mentioned in the following, and some references therein. When { } ∞ =1 has mean uncertainty, sample mean / probably cannot converge to a unique constant almost every- where (shortly a.e., which should be well defined) under a nonadditive probability or a set of probabilities. Marinacci [8], Teran [9], and some of the references therein investi- gate the SLLN via Choquet integrals related to completely monotone capacity . ey suppose that { } ∞ =1 is a sequence of independent and identically distributed random variables under capacity and prove that all the limit points of convergent subsequences of sample mean / belong to an interval [ [ 1 ], − [− 1 ]] with probability 1 (w.p. 1 for short) under ; that is, ( [ 1 ]≤ lim inf →∞ ≤ lim sup →∞ ≤ − [− 1 ] ) = 1. (1) Recently, Chen [10] and Chen et al. [11] prove the SLLN via a sublinear expectation E. ey suppose that { } ∞ =1 is a sequence of independent random variables under E (see Peng [4]) and prove that V ( ≤ lim inf →∞ ≤ lim sup →∞ ≤ ) = 1, (2) where V is the lower probability (see Halpern [12]) corre- sponding to E, := −E[− 1 ] and := E[ 1 ]. It is obvious Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 645947, 7 pages http://dx.doi.org/10.1155/2014/645947
Transcript of Research Article Weak and Strong Limit Theorems for...
Research ArticleWeak and Strong Limit Theorems for StochasticProcesses under Nonadditive Probability
Xiaoyan Chen1 and Zengjing Chen2
1 Graduate Department of Financial Engineering Ajou University Suwon 443-749 Republic of Korea2 School of Mathematics Shandong University Jinan 250100 China
Correspondence should be addressed to Xiaoyan Chen cxy 161977163com
Received 9 December 2013 Accepted 23 December 2013 Published 13 February 2014
Academic Editor Litan Yan
Copyright copy 2014 X Chen and Z Chen This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper extends laws of large numbers under upper probability to sequences of stochastic processes generated by linearinterpolation This extension characterizes the relation between sequences of stochastic processes and subsets of continuousfunction space in the framework of upper probability Limit results for sequences of functional random variables and some usefulinequalities are also obtained as applications
1 Introduction
Laws of large numbers are the cornerstones of theory ofprobability and statistics As we know under appropriateassumptions the well-known strong law of large numbers(SLLN for short) states that for a sequence of random vari-ables 119883
119899infin
119899=1 its sample mean 119878
119899119899 = sum
119899
119894=1119883119894119899 converges
to a unique constant almost surely in the framework ofprobability But many empirical analyses and theoreticalworks show us that nonadditive probability and nonlinearexpectation are very probably faced in economics financenumber theory statistics and many other fields such ascapacity Choquet integral (see Choquet [1]) (nontrivial)119892-probability (nontrivial) 119892-expectation (see El Karoui etal [2]) and 119866-expectation (see Peng [3]) And for eachnonadditive probability say 119888 we can define many differentexpectations related to 119888 denoted by 119864
119888 For nonlinear119864119888 random variables 119883
119899may have mean uncertainty that
is 119864119888[119883119899] = minus 119864
119888
[minus119883119899] or variance uncertainty that is
119864119888
[1198832
119899] = minus 119864
119888
[minus1198832
119899] In such cases there are many scholars
that investigate the limit theorems under 119864119888 or 119888 such as thelaws of large numbers laws of iterated logarithm central limittheorems under either 119864119888 or 119888 and other related problemsOne can refer to Peng [4 5] Chen and Hu [6] Wu andChen [7] the papers mentioned in the following and somereferences therein
When 119883119899infin
119899=1has mean uncertainty sample mean 119878
119899119899
probably cannot converge to a unique constant almost every-where (shortly ae which should be well defined) under anonadditive probability or a set of probabilities Marinacci[8] Teran [9] and some of the references therein investi-gate the SLLN via Choquet integrals related to completelymonotone capacity 119888They suppose that 119883
119899infin
119899=1is a sequence
of independent and identically distributed random variablesunder capacity 119888 and prove that all the limit points ofconvergent subsequences of sample mean 119878
119899119899 belong to
an interval [119864119888[1198831] minus119864119888
[minus1198831]] with probability 1 (wp 1 for
short) under 119888 that is
119888 (119864119888
[1198831] le lim inf119899rarrinfin
119878119899
119899le lim sup119899rarrinfin
119878119899
119899le minus119864119888
[minus1198831] ) = 1
(1)
Recently Chen [10] and Chen et al [11] prove the SLLNvia a sublinear expectation E They suppose that 119883
119899infin
119899=1is a
sequence of independent randomvariables underE (see Peng[4]) and prove that
V(120583 le lim inf119899rarrinfin
119878119899
119899le lim sup119899rarrinfin
119878119899
119899le 120583) = 1 (2)
where V is the lower probability (see Halpern [12]) corre-sponding to E 120583 = minusE[minus119883
1] and 120583 = E[119883
1] It is obvious
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 645947 7 pageshttpdxdoiorg1011552014645947
2 Abstract and Applied Analysis
that [120583 120583] is a subset of [119864119888[1198831] minus119864119888
[minus1198831]] On the other
hand Chen [10] and Chen [13] prove that any element of[120583 120583] is the limit of certain convergent subsequence of samplemean wp 1 under upper probability 119881 corresponding to E
This paper is motivated by the problem of limit theoremsof sequences of stochastic processes in the framework ofnonadditive probabilities and the estimation of expectationsof functionals of stock prices with ambiguity If there isno mean uncertainty they are trivial But if there is meanuncertainty then as the SLLN of random variables undernonadditive probability behaves limit theorems related tostochastic processes become interesting and different fromclassical case Chen [14] investigates a limit theorem for119866-quadratic variational process in the framework of 119866-expectation More generally for random variables 119883
119899infin
119899=1
with mean uncertainty in the framework of upper andlower probabilities (119881 V) we consider a simple sequence ofstochastic processes 120578
119899(119905) 119905 isin [0 1]
infin
119899=1generated by linearly
interpolating 119878119894119899 at 119894119899 X Chen and Z Chen [15] prove that
all the limit points of subsequences of 120578119899infin
119899=1are elements of
119869(120583 120583) wp 1 under lower probability V namely
V (Clust 120578119899 sub 119869 (120583 120583)) = 1 (3)
where 119869(120583 120583) is a subset of continuous function space on[0 1] (see Section 2) Conversely for any element of 119869(120583 120583)is it a limit point of certain subsequence of 120578
119899infin
119899=1wp 1
under upper probability119881 In otherwords does the followingstatement
119881 (119909 isin Clust 120578119899) = 1 (4)
hold trueIn this paper we will employ the independence condition
of Peng [4] to investigate this problem and prove thatunder certain conditions it holds true We will see that thisstrong form can be implied by a weak form (see Section 4)Under continuous upper probability our strong limit theorembecomes weaker than our weak one From the face of thismeaning it is different from classical framework But in factit coincides with the classical case We also extend our stronglimit theorem to functional random variables and show someuseful inequalities under continuous upper probability 119881
The remaining part of this paper is organized as followsIn Section 2 we recall some basic definitions and propertiesof lower and upper probabilities And we will also givebasic assumptions for all of the subsequent sections Someauxiliary lemmas are proved in Section 3 In Section 4 weprove a weak limit theorem under general upper probabilitySection 5 mainly presents a strong limit theorem undercontinuous upper probability and its extension to functionalrandom variables In Section 6 we give a simple example asapplications in finance
2 Basic Settings
LetΩ be a nonempty setF denotes a 120590-algebra of subsets ofΩ Let (119881 V) be a pair of nonadditive probabilities related toa set of probabilitiesP onmeasurable space (ΩF) given by
119881 (119860) = sup119876isinP
119876 (119860) V (119860) = inf119876isinP
119876 (119860) forall119860 isin F
(5)
It is obvious that upper probability 119881 and lower prob-ability V are conjugate capacities (see Choquet [1]) thatis (1) normalization 119881(Ω) = V(Ω) = 1 119881(0) = V(0) =
0 (2) monotonicity for all 119860 119861 isin F if 119860 sube 119861 then 119881(119860) le119881(119861) and V(119860) le V(119861) (3) conjugation for all119860 isin F V(119860) =1 minus 119881(119860
119888
) where 119860119888 denotes the complementary set of 119860Moreover we can easily get the following properties
which are useful in this paper (see also Chen et al [11])
Proposition 1 For any sequence of sets 119860119899isin F 119899 ge 1 we
have the following
(i) Subadditivity of 119881119881(suminfin119899=1
119860119899) le suminfin
119899=1119881(119860119899)
(ii) Lower continuity of 119881 if 119860119899
uarr 119860 then 119881(119860) =
lim119899rarrinfin
119881(119860119899)
(iii) Upper continuity of V if 119860119899
darr 119860 then V(119860) =
lim119899rarrinfin
V(119860119899)
(iv) If V(119860119899) = 1 for all 119899 ge 1 then V(⋂infin
119899=1119860119899) = 1
We say upper probability 119881 (resp lower probability V) iscontinuous if and only if it is upper and lower continuousObviously upper probability 119881 is continuous if and only iflower probability V is continuous
The corresponding pair of upper and lower expectations(EE) of (119881 V) is given as follows
E [119883] = sup119876isinP
119864119876[119883] E [119883] = inf
119876isinP119864119876[119883] forall119883 isin M
(6)
whereM denotes the set of all real-valued random variables119883 on (ΩF) such that sup
119876isinP119864119876[119883] lt infin Obviously E is asublinear expectation (see Peng [16])
Definition 2 (see Peng [16]) Let 119883119899infin
119899=1be a sequence of
random variables on (ΩF) inM We say it is a sequence ofindependent random variables under upper expectation E iffor all real-valued continuous functions 120593 on R119899 denoted by120593 isin 119862(R119899) with linear growth condition that is there existsa constant 119862 gt 0 st
1003816100381610038161003816120593 (119909)1003816100381610038161003816 le 119862 (1 + |119909|) forall119909 isin R
119899
(7)
we have
E [120593 (1198831 1198832 119883
119899)]
= E [E[120593 (119910119883119899)]119910=(119883
11198832119883119899minus1)] forall119899 ge 2
(8)
Abstract and Applied Analysis 3
Throughout this paper we assume (unless otherwisespecified) that 119883
119899infin
119899=1is a sequence of independent random
variables under upper expectation E satisfying
E [119883119899] = 120583 E [119883
119899] = 120583 E[sup
119899ge1
100381610038161003816100381611988311989910038161003816100381610038162
] lt infin (9)
for all 119899 ge 1 respectively where minusinfin lt 120583 le 120583 lt infinSet 1198780
= 0 and 119878119899
= sum119899
119894=1119883119894for any 119899 ge 1 We
define a sequence of stochastic processes 120578119899infin
119899=1by linearly
interpolating 119878119894119899 at 119894119899 for each 119899 ge 1 and 1 le 119894 le 119899 that is
120578119899(119905) =
1
119899(1 + [119899119905] minus 119899119905) 119878
[119899119905]+1
119899(119899119905 minus [119899119905]) 119878
[119899119905]+1
forall119905 isin [0 1]
(10)
where [119909] denotes the greatest integer which is less or equalto a nonnegative number 119909
Let 119862[0 1] be a linear space of all real-valued continuousfunctions on [0 1] with supremum as its norm denoted bysdot Let 119869(120583 120583) be a subset of119862[0 1] such that all the functions119909 isin 119869(120583 120583) are absolutely continuous on [0 1] with 119909(0) = 0
and 120583 le 1199091015840
(119905) le 120583 almost everywhere on [0 1] Thus we caneasily have the following
Proposition 3 119869(120583 120583) is compact
3 Auxiliary Lemmas
Before investigating the convergence problem of sequence120578119899infin
119899=1under upper probability in this section we first give
some useful lemmas
Definition 4 A set119860 isin F is said to be a polar set if119881(119860) = 0We say an event holds quasisurely (qs for short) if it holdsoutside a polar set
We first give the following property
Lemma 5 The sequence 120578119899infin
119899=1of functions on [0 1] is
relatively compact wp 1 under lower probability V
Proof For each 119899 ge 1 function 120578119899can be rewritten as
120578119899(119905) =
119878119894minus1
119899+ (119878119894minus 119878119894minus1) (119905 minus
119894 minus 1
119899)
times 119868[(119894minus1)119899 119894119899)
(119905) +119878119899
1198991198681(119905) forall119905 isin [0 1]
(11)
Obviously for each 119899 ge 1 120578119899(0) = 0 and for any 1 le 119894 le
119899 the first-order derivative of 120578119899with respect to 119905 for every
120596 isin Ω is
1205781015840
119899(119905) = 119878
119894minus 119878119894minus1
= 119883119894 forall119905 isin (
119894 minus 1
119899119894
119899) (12)
Then the difference of 120578119899with respect to 119905 follows that for
any 119904 119905 isin [0 1] with 119904 le 119905
120578119899(119905) minus 120578
119899(119904) = int
119905
119904
1205781015840
(119903) 119889119903
= int[119899119905]119899
[119899119904]119899
1205781015840
(119903) 119889119903 + int119905
[119899119905]119899
1205781015840
(119903) 119889119903
minus int119904
[119899119904]119899
1205781015840
(119903) 119889119903
=119878[119899119905]
minus 119878[119899119904]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899)
minus 119883[119899119904]+1
(119904 minus[119899119904]
119899)
(13)
From E[sup119899ge1
|119883119899|2
] lt infin we have 119872 = sup119899ge1
|119883119899| lt
infin 119902119904 Thus we can get an upper bound of the norm of 120578119899
as follows
10038171003817100381710038171205781198991003817100381710038171003817 = sup119905isin[01]
1003816100381610038161003816120578119899 (119905)1003816100381610038161003816 le
sum119899
119894=1
10038161003816100381610038161198831198941003816100381610038161003816
119899le 119872 119902119904 (14)
In addition for any 119904 119905 isin [0 1] such that |119905 minus 119904| le 1119899 wecan get from (13) that
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816 le 119872 |119905 minus 119904| 119902119904 (15)
In fact without loss of generality we assume that 119904 le 119905 if[119899119905] gt 119899119904 thus [119899119905] = [119899119904] + 1 then from (13) it follows that1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)
1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816
119883[119899119905]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]
([119899119905]
119899minus 119904)+119883
[119899119905]+1(119905minus
[119899119905]
119899)10038161003816100381610038161003816100381610038161003816le119872 (119905 minus 119904) 119902119904
(16)
Otherwise if [119899119905] le 119899119904 thus [119899119904] le [119899119905] lt [119899119904] + 1 le
[119899119905] + 1 which implies that [119899119905] = [119899119904] then from (13) wehave
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=1003816100381610038161003816119883[119899119905]+1
1003816100381610038161003816 (119905 minus 119904) le 119872 (119905 minus 119904) 119902119904
(17)
Hence from (16) and (17) we know that (15) holds trueThus we can easily get that 120578
119899infin
119899=1is equicontinuous with
respect to 119905wp 1 under lower probability V from property (iv)of Proposition 1 Together with (14) this sequence 120578
119899infin
119899=1is
relatively compact in119862[0 1]wp 1 under VWe get the desiredresult
The following lemma is very useful in the proofs of ourmain theorems and its proof is similar as Theorem 31 of Hu[17] Here we omit its proof
4 Abstract and Applied Analysis
Lemma 6 Given a sequence of independent random variables119884119899infin
119899=1underE we assume that there exist two constants 119886 lt 119887
such that E[119884119899] = 119886 and E[119884
119899] = 119887 for all 119899 ge 1 and we
also assume that sup119899ge1
E[|119884119899|2
] lt infin Then for any increasingsubsequence 119899
119896infin
119896=1of N satisfying 119899
119896minus 119899119896minus1
converges to infinas 119899 tends toinfin and for any 120593 isin 119862(119877) with linear growth wehave
lim119896rarrinfin
E [120593(119878119899119896
minus 119878119899119896minus1
119899119896minus 119899119896minus1
)] = sup119886le119906le119887
120593 (119906) (18)
where 119878119898= sum119898
119894=1119884119894for all119898 ge 1
4 Weak Limit Theorem
In this section we will investigate the weak convergenceproblem of 120578
119899infin
119899=1under general upper probability
Theorem 7 For any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists asubsequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598) = 1 (19)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598
Proof For any 119909 isin 119869(120583 120583) and 120598 gt 0 by Lemma 5 we onlyneed to find a subsequence 120578
119899119898
infin
119898=1satisfying (19) Set
119860119898= 120596 isin Ω
10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598 forall119898 ge 1 (20)
Note that for any integer 119897 ge 1
119881 (119860119898)
= 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)
+ 119909(119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
ge 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)1003816100381610038161003816100381610038161003816
+ sup119905isin[(119894minus1)119897 119894119897]1le119894le119897
1003816100381610038161003816100381610038161003816119909 (
119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
(21)
Denoting119863 = max|120583| |120583| since 119909 isin 119869(120583 120583) thus for all1 le 119894 le 119897 |119909((119894 minus 1)119897) minus 119909(119905)| le 119863119897 for all 119905 isin [(119894 minus 1)119897 119894119897]Hence taking 119897 ge 3119863120598 we have
119881 (119860119898) ge 119881(
1003816100381610038161003816100381610038161003816120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897)
minus 119909(119894 minus 1
119897) minus 119909 (
119894 minus 2
119897)1003816100381610038161003816100381610038161003816le
120598
3119897 2 le 119894 le 119897)
(22)
Let 119899119898119897 be a positive integer for any 119898 ge 1 then by the
definition of 120578119899119898
(see (10)) it follows that for 2 le 119894 le 119897 119897 ge3119863120598 and119898 ge 1
120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897) =
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898
(23)
In addition since 119909 isin 119869(120583 120583) we know that
119886119894
119897≜ 119909 (
119894 minus 1
119897) minus 119909 (
119894 minus 2
119897) isin [
120583
119897120583
119897]
forall2 le 119894 le 119897 119897 ge3119863
120598
(24)
Then it follows that
119881 (119860119898) ge 119881(
119897
⋂119894=2
100381610038161003816100381610038161003816100381610038161003816
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894
100381610038161003816100381610038161003816100381610038161003816le120598
3)
forall119898 ge 1
(25)
For 2 le 119894 le 119897 and 120575 isin (0 1205983) we set
119892120575
(119910) =
1 119910 isin [minus120598
3+ 120575
120598
3minus 120575]
119910 + (1205983)
120575 119910 isin (minus
120598
3 minus
120598
3+ 120575)
(1205983) minus 120575 minus 119910
120575 119910 isin (
120598
3minus 120575
120598
3)
0 119910 isin [120598
3 +infin) cup (minusinfin minus
120598
3]
(26)
Obviously Π119897119894=2119892120575
(119910119894) is a continuous function on R119897minus1
satisfying linear growth condition Since 119883119899infin
119899=1is indepen-
dent under E (see Definition 2) from (25) we have
119881 (119860119898) ge E[
119897
prod119894=2
119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
=
119897
prod119894=2
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
(27)
Abstract and Applied Analysis 5
Since for all 2 le 119894 le 119897 with 119897 ge 3119863120598 and 119899 ge 1 E[119883119899minus
119886119894] = 120583minus119886
119894E[119883
119899minus119886119894] = 120583minus119886
119894and sup
119899ge1E[|119883119899minus119886119894|2
] lt infinlet 119899119898tend toinfin as119898 tends toinfin then by Lemma 6 we have
lim119898rarrinfin
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
= sup120583minus119886119894le119906le120583minus119886
119894
119892120575
(119906) = 1
(28)
since 119886119894isin [120583 120583] for 2 le 119894 le 119897 Thus from (27) and (28) it fol-
lows that lim inf119898rarrinfin
119881(119860119898) ge 1 Obviously 119881(119860
119898) le 1 for
all119898 ge 1 Hence this theorem follows
Corollary 8 Let 120593 be a real-valued continuous functional on119862[0 1] then for any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists a sub-sequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038161003816100381610038161003816120593 (120578119899119898
) minus 120593 (119909)10038161003816100381610038161003816le 120598) = 1 (29)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598In particular if we assume that 120593(119909) = 119909(1) for all 119909 isin
119862[0 1] then we have
lim119898rarrinfin
119881(
100381610038161003816100381610038161003816100381610038161003816
119878119899119898
119899119898
minus 119909 (1)
100381610038161003816100381610038161003816100381610038161003816le 120598) = 1 (30)
where 119909(1) isin [120583 120583]
5 Strong Limit Theorem underContinuous Upper Probability
In the previous Sections 2ndash4 we consider the general upperprobability 119881 For the sake of technique in this section wefurther assume that 119881 is continuous and investigate a stronglimit theorem of 120578
119899infin
119899=1under such a continuous upper
probability 119881 and its extension
51 Strong Limit Theorem
Theorem 9 Any 119909 isin 119869(120583 120583) is a limit point of somesubsequence of 120578
119899infin
119899=1wp 1 under 119881 that is
119881 (119909 isin Clust 120578119899) = 1 (31)
where Clust119909119899 denotes the cluster set of all the limit points of
real sequence 119909119899infin
119899=1
Proof From Lemma 5 since 119881 is continuous we only needto prove that for any 119909 isin 119869(120583 120583) and any 120598 gt 0
119881(lim inf119899rarrinfin
1003817100381710038171003817120578119899 minus 1199091003817100381710038171003817 le 120598)
= 119881(
infin
⋂119899=1
infin
⋃119898=119899
1003817100381710038171003817120578119898 minus 119909
1003817100381710038171003817 le 120598) = 1
(32)
Let 119860119898infin
119898=1and 119863 be defined the same as in the proof
of Theorem 7 Then it is sufficient to prove that for any fixed120598 gt 0 we can find a subsequence 119899
119898infin
119898=1of N such that
119881(lim inf119898rarrinfin
120578119899119898
minus 119909 le 120598) = 119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = 1 (33)
Take 119899119898= 119897119898 for 119898 ge 1 where 119897 ge 3119863120598 is an integer
FromTheorem 7 and the continuity of 119881 we can get
119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = lim119898rarrinfin
119881(
infin
⋃119895=119898
119860119895) ge lim119898rarrinfin
119881 (119860119898) = 1
(34)
Thus this theorem is proved
Remark 10 From the proof of Theorem 9 we can see thatit is implied by weak limit Theorem 7 under continuousupper probability It seems that ldquoweak limit theoremrdquo isstronger than ldquostrong limit theoremrdquo under continuous upperprobability IfP is a singleton thus we have 120583 = 120583 Then ourldquostrong limit theoremrdquo is not the same form as the strong lawof large numbers for sequences of random variables since theformer form is related to inferior limit and the latter one isrelated to limit
52 Extension to Functional Random Variables ByTheorem 9 we can easily get the following limit resultfor functional random variables
Corollary 11 Let 120593 be a real-valued continuous functionaldefined on 119862[0 1] then we have for any 119909 isin 119869(120583 120583)
119881 (120593 (119909) isin Clust 120593 (120578119899)) = 1 (35)
In particular
119881( sup119909isin119869(120583120583)
120593 (119909) le lim sup119899rarrinfin
120593 (120578119899))
= 119881( inf119909isin119869(120583120583)
120593 (119909) ge lim inf119899rarrinfin
120593 (120578119899)) = 1
(36)
From the proof ofTheorem 31 and Corollary 32 of Chenet al [11] the following lemma can be easily obtained
Lemma 12 Supposing 119891 is a real-valued continuous functionon R then
V( inf119910isin[120583120583]
119891 (119909) le lim inf119899rarrinfin
119891(119878119899
119899)
le lim sup119899rarrinfin
119891(119878119899
119899) le sup119910isin[120583120583]
119891 (119909)) = 1
(37)
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
that [120583 120583] is a subset of [119864119888[1198831] minus119864119888
[minus1198831]] On the other
hand Chen [10] and Chen [13] prove that any element of[120583 120583] is the limit of certain convergent subsequence of samplemean wp 1 under upper probability 119881 corresponding to E
This paper is motivated by the problem of limit theoremsof sequences of stochastic processes in the framework ofnonadditive probabilities and the estimation of expectationsof functionals of stock prices with ambiguity If there isno mean uncertainty they are trivial But if there is meanuncertainty then as the SLLN of random variables undernonadditive probability behaves limit theorems related tostochastic processes become interesting and different fromclassical case Chen [14] investigates a limit theorem for119866-quadratic variational process in the framework of 119866-expectation More generally for random variables 119883
119899infin
119899=1
with mean uncertainty in the framework of upper andlower probabilities (119881 V) we consider a simple sequence ofstochastic processes 120578
119899(119905) 119905 isin [0 1]
infin
119899=1generated by linearly
interpolating 119878119894119899 at 119894119899 X Chen and Z Chen [15] prove that
all the limit points of subsequences of 120578119899infin
119899=1are elements of
119869(120583 120583) wp 1 under lower probability V namely
V (Clust 120578119899 sub 119869 (120583 120583)) = 1 (3)
where 119869(120583 120583) is a subset of continuous function space on[0 1] (see Section 2) Conversely for any element of 119869(120583 120583)is it a limit point of certain subsequence of 120578
119899infin
119899=1wp 1
under upper probability119881 In otherwords does the followingstatement
119881 (119909 isin Clust 120578119899) = 1 (4)
hold trueIn this paper we will employ the independence condition
of Peng [4] to investigate this problem and prove thatunder certain conditions it holds true We will see that thisstrong form can be implied by a weak form (see Section 4)Under continuous upper probability our strong limit theorembecomes weaker than our weak one From the face of thismeaning it is different from classical framework But in factit coincides with the classical case We also extend our stronglimit theorem to functional random variables and show someuseful inequalities under continuous upper probability 119881
The remaining part of this paper is organized as followsIn Section 2 we recall some basic definitions and propertiesof lower and upper probabilities And we will also givebasic assumptions for all of the subsequent sections Someauxiliary lemmas are proved in Section 3 In Section 4 weprove a weak limit theorem under general upper probabilitySection 5 mainly presents a strong limit theorem undercontinuous upper probability and its extension to functionalrandom variables In Section 6 we give a simple example asapplications in finance
2 Basic Settings
LetΩ be a nonempty setF denotes a 120590-algebra of subsets ofΩ Let (119881 V) be a pair of nonadditive probabilities related toa set of probabilitiesP onmeasurable space (ΩF) given by
119881 (119860) = sup119876isinP
119876 (119860) V (119860) = inf119876isinP
119876 (119860) forall119860 isin F
(5)
It is obvious that upper probability 119881 and lower prob-ability V are conjugate capacities (see Choquet [1]) thatis (1) normalization 119881(Ω) = V(Ω) = 1 119881(0) = V(0) =
0 (2) monotonicity for all 119860 119861 isin F if 119860 sube 119861 then 119881(119860) le119881(119861) and V(119860) le V(119861) (3) conjugation for all119860 isin F V(119860) =1 minus 119881(119860
119888
) where 119860119888 denotes the complementary set of 119860Moreover we can easily get the following properties
which are useful in this paper (see also Chen et al [11])
Proposition 1 For any sequence of sets 119860119899isin F 119899 ge 1 we
have the following
(i) Subadditivity of 119881119881(suminfin119899=1
119860119899) le suminfin
119899=1119881(119860119899)
(ii) Lower continuity of 119881 if 119860119899
uarr 119860 then 119881(119860) =
lim119899rarrinfin
119881(119860119899)
(iii) Upper continuity of V if 119860119899
darr 119860 then V(119860) =
lim119899rarrinfin
V(119860119899)
(iv) If V(119860119899) = 1 for all 119899 ge 1 then V(⋂infin
119899=1119860119899) = 1
We say upper probability 119881 (resp lower probability V) iscontinuous if and only if it is upper and lower continuousObviously upper probability 119881 is continuous if and only iflower probability V is continuous
The corresponding pair of upper and lower expectations(EE) of (119881 V) is given as follows
E [119883] = sup119876isinP
119864119876[119883] E [119883] = inf
119876isinP119864119876[119883] forall119883 isin M
(6)
whereM denotes the set of all real-valued random variables119883 on (ΩF) such that sup
119876isinP119864119876[119883] lt infin Obviously E is asublinear expectation (see Peng [16])
Definition 2 (see Peng [16]) Let 119883119899infin
119899=1be a sequence of
random variables on (ΩF) inM We say it is a sequence ofindependent random variables under upper expectation E iffor all real-valued continuous functions 120593 on R119899 denoted by120593 isin 119862(R119899) with linear growth condition that is there existsa constant 119862 gt 0 st
1003816100381610038161003816120593 (119909)1003816100381610038161003816 le 119862 (1 + |119909|) forall119909 isin R
119899
(7)
we have
E [120593 (1198831 1198832 119883
119899)]
= E [E[120593 (119910119883119899)]119910=(119883
11198832119883119899minus1)] forall119899 ge 2
(8)
Abstract and Applied Analysis 3
Throughout this paper we assume (unless otherwisespecified) that 119883
119899infin
119899=1is a sequence of independent random
variables under upper expectation E satisfying
E [119883119899] = 120583 E [119883
119899] = 120583 E[sup
119899ge1
100381610038161003816100381611988311989910038161003816100381610038162
] lt infin (9)
for all 119899 ge 1 respectively where minusinfin lt 120583 le 120583 lt infinSet 1198780
= 0 and 119878119899
= sum119899
119894=1119883119894for any 119899 ge 1 We
define a sequence of stochastic processes 120578119899infin
119899=1by linearly
interpolating 119878119894119899 at 119894119899 for each 119899 ge 1 and 1 le 119894 le 119899 that is
120578119899(119905) =
1
119899(1 + [119899119905] minus 119899119905) 119878
[119899119905]+1
119899(119899119905 minus [119899119905]) 119878
[119899119905]+1
forall119905 isin [0 1]
(10)
where [119909] denotes the greatest integer which is less or equalto a nonnegative number 119909
Let 119862[0 1] be a linear space of all real-valued continuousfunctions on [0 1] with supremum as its norm denoted bysdot Let 119869(120583 120583) be a subset of119862[0 1] such that all the functions119909 isin 119869(120583 120583) are absolutely continuous on [0 1] with 119909(0) = 0
and 120583 le 1199091015840
(119905) le 120583 almost everywhere on [0 1] Thus we caneasily have the following
Proposition 3 119869(120583 120583) is compact
3 Auxiliary Lemmas
Before investigating the convergence problem of sequence120578119899infin
119899=1under upper probability in this section we first give
some useful lemmas
Definition 4 A set119860 isin F is said to be a polar set if119881(119860) = 0We say an event holds quasisurely (qs for short) if it holdsoutside a polar set
We first give the following property
Lemma 5 The sequence 120578119899infin
119899=1of functions on [0 1] is
relatively compact wp 1 under lower probability V
Proof For each 119899 ge 1 function 120578119899can be rewritten as
120578119899(119905) =
119878119894minus1
119899+ (119878119894minus 119878119894minus1) (119905 minus
119894 minus 1
119899)
times 119868[(119894minus1)119899 119894119899)
(119905) +119878119899
1198991198681(119905) forall119905 isin [0 1]
(11)
Obviously for each 119899 ge 1 120578119899(0) = 0 and for any 1 le 119894 le
119899 the first-order derivative of 120578119899with respect to 119905 for every
120596 isin Ω is
1205781015840
119899(119905) = 119878
119894minus 119878119894minus1
= 119883119894 forall119905 isin (
119894 minus 1
119899119894
119899) (12)
Then the difference of 120578119899with respect to 119905 follows that for
any 119904 119905 isin [0 1] with 119904 le 119905
120578119899(119905) minus 120578
119899(119904) = int
119905
119904
1205781015840
(119903) 119889119903
= int[119899119905]119899
[119899119904]119899
1205781015840
(119903) 119889119903 + int119905
[119899119905]119899
1205781015840
(119903) 119889119903
minus int119904
[119899119904]119899
1205781015840
(119903) 119889119903
=119878[119899119905]
minus 119878[119899119904]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899)
minus 119883[119899119904]+1
(119904 minus[119899119904]
119899)
(13)
From E[sup119899ge1
|119883119899|2
] lt infin we have 119872 = sup119899ge1
|119883119899| lt
infin 119902119904 Thus we can get an upper bound of the norm of 120578119899
as follows
10038171003817100381710038171205781198991003817100381710038171003817 = sup119905isin[01]
1003816100381610038161003816120578119899 (119905)1003816100381610038161003816 le
sum119899
119894=1
10038161003816100381610038161198831198941003816100381610038161003816
119899le 119872 119902119904 (14)
In addition for any 119904 119905 isin [0 1] such that |119905 minus 119904| le 1119899 wecan get from (13) that
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816 le 119872 |119905 minus 119904| 119902119904 (15)
In fact without loss of generality we assume that 119904 le 119905 if[119899119905] gt 119899119904 thus [119899119905] = [119899119904] + 1 then from (13) it follows that1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)
1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816
119883[119899119905]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]
([119899119905]
119899minus 119904)+119883
[119899119905]+1(119905minus
[119899119905]
119899)10038161003816100381610038161003816100381610038161003816le119872 (119905 minus 119904) 119902119904
(16)
Otherwise if [119899119905] le 119899119904 thus [119899119904] le [119899119905] lt [119899119904] + 1 le
[119899119905] + 1 which implies that [119899119905] = [119899119904] then from (13) wehave
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=1003816100381610038161003816119883[119899119905]+1
1003816100381610038161003816 (119905 minus 119904) le 119872 (119905 minus 119904) 119902119904
(17)
Hence from (16) and (17) we know that (15) holds trueThus we can easily get that 120578
119899infin
119899=1is equicontinuous with
respect to 119905wp 1 under lower probability V from property (iv)of Proposition 1 Together with (14) this sequence 120578
119899infin
119899=1is
relatively compact in119862[0 1]wp 1 under VWe get the desiredresult
The following lemma is very useful in the proofs of ourmain theorems and its proof is similar as Theorem 31 of Hu[17] Here we omit its proof
4 Abstract and Applied Analysis
Lemma 6 Given a sequence of independent random variables119884119899infin
119899=1underE we assume that there exist two constants 119886 lt 119887
such that E[119884119899] = 119886 and E[119884
119899] = 119887 for all 119899 ge 1 and we
also assume that sup119899ge1
E[|119884119899|2
] lt infin Then for any increasingsubsequence 119899
119896infin
119896=1of N satisfying 119899
119896minus 119899119896minus1
converges to infinas 119899 tends toinfin and for any 120593 isin 119862(119877) with linear growth wehave
lim119896rarrinfin
E [120593(119878119899119896
minus 119878119899119896minus1
119899119896minus 119899119896minus1
)] = sup119886le119906le119887
120593 (119906) (18)
where 119878119898= sum119898
119894=1119884119894for all119898 ge 1
4 Weak Limit Theorem
In this section we will investigate the weak convergenceproblem of 120578
119899infin
119899=1under general upper probability
Theorem 7 For any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists asubsequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598) = 1 (19)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598
Proof For any 119909 isin 119869(120583 120583) and 120598 gt 0 by Lemma 5 we onlyneed to find a subsequence 120578
119899119898
infin
119898=1satisfying (19) Set
119860119898= 120596 isin Ω
10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598 forall119898 ge 1 (20)
Note that for any integer 119897 ge 1
119881 (119860119898)
= 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)
+ 119909(119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
ge 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)1003816100381610038161003816100381610038161003816
+ sup119905isin[(119894minus1)119897 119894119897]1le119894le119897
1003816100381610038161003816100381610038161003816119909 (
119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
(21)
Denoting119863 = max|120583| |120583| since 119909 isin 119869(120583 120583) thus for all1 le 119894 le 119897 |119909((119894 minus 1)119897) minus 119909(119905)| le 119863119897 for all 119905 isin [(119894 minus 1)119897 119894119897]Hence taking 119897 ge 3119863120598 we have
119881 (119860119898) ge 119881(
1003816100381610038161003816100381610038161003816120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897)
minus 119909(119894 minus 1
119897) minus 119909 (
119894 minus 2
119897)1003816100381610038161003816100381610038161003816le
120598
3119897 2 le 119894 le 119897)
(22)
Let 119899119898119897 be a positive integer for any 119898 ge 1 then by the
definition of 120578119899119898
(see (10)) it follows that for 2 le 119894 le 119897 119897 ge3119863120598 and119898 ge 1
120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897) =
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898
(23)
In addition since 119909 isin 119869(120583 120583) we know that
119886119894
119897≜ 119909 (
119894 minus 1
119897) minus 119909 (
119894 minus 2
119897) isin [
120583
119897120583
119897]
forall2 le 119894 le 119897 119897 ge3119863
120598
(24)
Then it follows that
119881 (119860119898) ge 119881(
119897
⋂119894=2
100381610038161003816100381610038161003816100381610038161003816
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894
100381610038161003816100381610038161003816100381610038161003816le120598
3)
forall119898 ge 1
(25)
For 2 le 119894 le 119897 and 120575 isin (0 1205983) we set
119892120575
(119910) =
1 119910 isin [minus120598
3+ 120575
120598
3minus 120575]
119910 + (1205983)
120575 119910 isin (minus
120598
3 minus
120598
3+ 120575)
(1205983) minus 120575 minus 119910
120575 119910 isin (
120598
3minus 120575
120598
3)
0 119910 isin [120598
3 +infin) cup (minusinfin minus
120598
3]
(26)
Obviously Π119897119894=2119892120575
(119910119894) is a continuous function on R119897minus1
satisfying linear growth condition Since 119883119899infin
119899=1is indepen-
dent under E (see Definition 2) from (25) we have
119881 (119860119898) ge E[
119897
prod119894=2
119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
=
119897
prod119894=2
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
(27)
Abstract and Applied Analysis 5
Since for all 2 le 119894 le 119897 with 119897 ge 3119863120598 and 119899 ge 1 E[119883119899minus
119886119894] = 120583minus119886
119894E[119883
119899minus119886119894] = 120583minus119886
119894and sup
119899ge1E[|119883119899minus119886119894|2
] lt infinlet 119899119898tend toinfin as119898 tends toinfin then by Lemma 6 we have
lim119898rarrinfin
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
= sup120583minus119886119894le119906le120583minus119886
119894
119892120575
(119906) = 1
(28)
since 119886119894isin [120583 120583] for 2 le 119894 le 119897 Thus from (27) and (28) it fol-
lows that lim inf119898rarrinfin
119881(119860119898) ge 1 Obviously 119881(119860
119898) le 1 for
all119898 ge 1 Hence this theorem follows
Corollary 8 Let 120593 be a real-valued continuous functional on119862[0 1] then for any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists a sub-sequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038161003816100381610038161003816120593 (120578119899119898
) minus 120593 (119909)10038161003816100381610038161003816le 120598) = 1 (29)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598In particular if we assume that 120593(119909) = 119909(1) for all 119909 isin
119862[0 1] then we have
lim119898rarrinfin
119881(
100381610038161003816100381610038161003816100381610038161003816
119878119899119898
119899119898
minus 119909 (1)
100381610038161003816100381610038161003816100381610038161003816le 120598) = 1 (30)
where 119909(1) isin [120583 120583]
5 Strong Limit Theorem underContinuous Upper Probability
In the previous Sections 2ndash4 we consider the general upperprobability 119881 For the sake of technique in this section wefurther assume that 119881 is continuous and investigate a stronglimit theorem of 120578
119899infin
119899=1under such a continuous upper
probability 119881 and its extension
51 Strong Limit Theorem
Theorem 9 Any 119909 isin 119869(120583 120583) is a limit point of somesubsequence of 120578
119899infin
119899=1wp 1 under 119881 that is
119881 (119909 isin Clust 120578119899) = 1 (31)
where Clust119909119899 denotes the cluster set of all the limit points of
real sequence 119909119899infin
119899=1
Proof From Lemma 5 since 119881 is continuous we only needto prove that for any 119909 isin 119869(120583 120583) and any 120598 gt 0
119881(lim inf119899rarrinfin
1003817100381710038171003817120578119899 minus 1199091003817100381710038171003817 le 120598)
= 119881(
infin
⋂119899=1
infin
⋃119898=119899
1003817100381710038171003817120578119898 minus 119909
1003817100381710038171003817 le 120598) = 1
(32)
Let 119860119898infin
119898=1and 119863 be defined the same as in the proof
of Theorem 7 Then it is sufficient to prove that for any fixed120598 gt 0 we can find a subsequence 119899
119898infin
119898=1of N such that
119881(lim inf119898rarrinfin
120578119899119898
minus 119909 le 120598) = 119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = 1 (33)
Take 119899119898= 119897119898 for 119898 ge 1 where 119897 ge 3119863120598 is an integer
FromTheorem 7 and the continuity of 119881 we can get
119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = lim119898rarrinfin
119881(
infin
⋃119895=119898
119860119895) ge lim119898rarrinfin
119881 (119860119898) = 1
(34)
Thus this theorem is proved
Remark 10 From the proof of Theorem 9 we can see thatit is implied by weak limit Theorem 7 under continuousupper probability It seems that ldquoweak limit theoremrdquo isstronger than ldquostrong limit theoremrdquo under continuous upperprobability IfP is a singleton thus we have 120583 = 120583 Then ourldquostrong limit theoremrdquo is not the same form as the strong lawof large numbers for sequences of random variables since theformer form is related to inferior limit and the latter one isrelated to limit
52 Extension to Functional Random Variables ByTheorem 9 we can easily get the following limit resultfor functional random variables
Corollary 11 Let 120593 be a real-valued continuous functionaldefined on 119862[0 1] then we have for any 119909 isin 119869(120583 120583)
119881 (120593 (119909) isin Clust 120593 (120578119899)) = 1 (35)
In particular
119881( sup119909isin119869(120583120583)
120593 (119909) le lim sup119899rarrinfin
120593 (120578119899))
= 119881( inf119909isin119869(120583120583)
120593 (119909) ge lim inf119899rarrinfin
120593 (120578119899)) = 1
(36)
From the proof ofTheorem 31 and Corollary 32 of Chenet al [11] the following lemma can be easily obtained
Lemma 12 Supposing 119891 is a real-valued continuous functionon R then
V( inf119910isin[120583120583]
119891 (119909) le lim inf119899rarrinfin
119891(119878119899
119899)
le lim sup119899rarrinfin
119891(119878119899
119899) le sup119910isin[120583120583]
119891 (119909)) = 1
(37)
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Throughout this paper we assume (unless otherwisespecified) that 119883
119899infin
119899=1is a sequence of independent random
variables under upper expectation E satisfying
E [119883119899] = 120583 E [119883
119899] = 120583 E[sup
119899ge1
100381610038161003816100381611988311989910038161003816100381610038162
] lt infin (9)
for all 119899 ge 1 respectively where minusinfin lt 120583 le 120583 lt infinSet 1198780
= 0 and 119878119899
= sum119899
119894=1119883119894for any 119899 ge 1 We
define a sequence of stochastic processes 120578119899infin
119899=1by linearly
interpolating 119878119894119899 at 119894119899 for each 119899 ge 1 and 1 le 119894 le 119899 that is
120578119899(119905) =
1
119899(1 + [119899119905] minus 119899119905) 119878
[119899119905]+1
119899(119899119905 minus [119899119905]) 119878
[119899119905]+1
forall119905 isin [0 1]
(10)
where [119909] denotes the greatest integer which is less or equalto a nonnegative number 119909
Let 119862[0 1] be a linear space of all real-valued continuousfunctions on [0 1] with supremum as its norm denoted bysdot Let 119869(120583 120583) be a subset of119862[0 1] such that all the functions119909 isin 119869(120583 120583) are absolutely continuous on [0 1] with 119909(0) = 0
and 120583 le 1199091015840
(119905) le 120583 almost everywhere on [0 1] Thus we caneasily have the following
Proposition 3 119869(120583 120583) is compact
3 Auxiliary Lemmas
Before investigating the convergence problem of sequence120578119899infin
119899=1under upper probability in this section we first give
some useful lemmas
Definition 4 A set119860 isin F is said to be a polar set if119881(119860) = 0We say an event holds quasisurely (qs for short) if it holdsoutside a polar set
We first give the following property
Lemma 5 The sequence 120578119899infin
119899=1of functions on [0 1] is
relatively compact wp 1 under lower probability V
Proof For each 119899 ge 1 function 120578119899can be rewritten as
120578119899(119905) =
119878119894minus1
119899+ (119878119894minus 119878119894minus1) (119905 minus
119894 minus 1
119899)
times 119868[(119894minus1)119899 119894119899)
(119905) +119878119899
1198991198681(119905) forall119905 isin [0 1]
(11)
Obviously for each 119899 ge 1 120578119899(0) = 0 and for any 1 le 119894 le
119899 the first-order derivative of 120578119899with respect to 119905 for every
120596 isin Ω is
1205781015840
119899(119905) = 119878
119894minus 119878119894minus1
= 119883119894 forall119905 isin (
119894 minus 1
119899119894
119899) (12)
Then the difference of 120578119899with respect to 119905 follows that for
any 119904 119905 isin [0 1] with 119904 le 119905
120578119899(119905) minus 120578
119899(119904) = int
119905
119904
1205781015840
(119903) 119889119903
= int[119899119905]119899
[119899119904]119899
1205781015840
(119903) 119889119903 + int119905
[119899119905]119899
1205781015840
(119903) 119889119903
minus int119904
[119899119904]119899
1205781015840
(119903) 119889119903
=119878[119899119905]
minus 119878[119899119904]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899)
minus 119883[119899119904]+1
(119904 minus[119899119904]
119899)
(13)
From E[sup119899ge1
|119883119899|2
] lt infin we have 119872 = sup119899ge1
|119883119899| lt
infin 119902119904 Thus we can get an upper bound of the norm of 120578119899
as follows
10038171003817100381710038171205781198991003817100381710038171003817 = sup119905isin[01]
1003816100381610038161003816120578119899 (119905)1003816100381610038161003816 le
sum119899
119894=1
10038161003816100381610038161198831198941003816100381610038161003816
119899le 119872 119902119904 (14)
In addition for any 119904 119905 isin [0 1] such that |119905 minus 119904| le 1119899 wecan get from (13) that
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816 le 119872 |119905 minus 119904| 119902119904 (15)
In fact without loss of generality we assume that 119904 le 119905 if[119899119905] gt 119899119904 thus [119899119905] = [119899119904] + 1 then from (13) it follows that1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)
1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816
119883[119899119905]
119899+ 119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]
([119899119905]
119899minus 119904)+119883
[119899119905]+1(119905minus
[119899119905]
119899)10038161003816100381610038161003816100381610038161003816le119872 (119905 minus 119904) 119902119904
(16)
Otherwise if [119899119905] le 119899119904 thus [119899119904] le [119899119905] lt [119899119904] + 1 le
[119899119905] + 1 which implies that [119899119905] = [119899119904] then from (13) wehave
1003816100381610038161003816120578119899 (119905) minus 120578119899 (119904)1003816100381610038161003816
=10038161003816100381610038161003816100381610038161003816119883[119899119905]+1
(119905 minus[119899119905]
119899) minus 119883
[119899119904]+1(119904 minus
[119899119904]
119899)10038161003816100381610038161003816100381610038161003816
=1003816100381610038161003816119883[119899119905]+1
1003816100381610038161003816 (119905 minus 119904) le 119872 (119905 minus 119904) 119902119904
(17)
Hence from (16) and (17) we know that (15) holds trueThus we can easily get that 120578
119899infin
119899=1is equicontinuous with
respect to 119905wp 1 under lower probability V from property (iv)of Proposition 1 Together with (14) this sequence 120578
119899infin
119899=1is
relatively compact in119862[0 1]wp 1 under VWe get the desiredresult
The following lemma is very useful in the proofs of ourmain theorems and its proof is similar as Theorem 31 of Hu[17] Here we omit its proof
4 Abstract and Applied Analysis
Lemma 6 Given a sequence of independent random variables119884119899infin
119899=1underE we assume that there exist two constants 119886 lt 119887
such that E[119884119899] = 119886 and E[119884
119899] = 119887 for all 119899 ge 1 and we
also assume that sup119899ge1
E[|119884119899|2
] lt infin Then for any increasingsubsequence 119899
119896infin
119896=1of N satisfying 119899
119896minus 119899119896minus1
converges to infinas 119899 tends toinfin and for any 120593 isin 119862(119877) with linear growth wehave
lim119896rarrinfin
E [120593(119878119899119896
minus 119878119899119896minus1
119899119896minus 119899119896minus1
)] = sup119886le119906le119887
120593 (119906) (18)
where 119878119898= sum119898
119894=1119884119894for all119898 ge 1
4 Weak Limit Theorem
In this section we will investigate the weak convergenceproblem of 120578
119899infin
119899=1under general upper probability
Theorem 7 For any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists asubsequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598) = 1 (19)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598
Proof For any 119909 isin 119869(120583 120583) and 120598 gt 0 by Lemma 5 we onlyneed to find a subsequence 120578
119899119898
infin
119898=1satisfying (19) Set
119860119898= 120596 isin Ω
10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598 forall119898 ge 1 (20)
Note that for any integer 119897 ge 1
119881 (119860119898)
= 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)
+ 119909(119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
ge 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)1003816100381610038161003816100381610038161003816
+ sup119905isin[(119894minus1)119897 119894119897]1le119894le119897
1003816100381610038161003816100381610038161003816119909 (
119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
(21)
Denoting119863 = max|120583| |120583| since 119909 isin 119869(120583 120583) thus for all1 le 119894 le 119897 |119909((119894 minus 1)119897) minus 119909(119905)| le 119863119897 for all 119905 isin [(119894 minus 1)119897 119894119897]Hence taking 119897 ge 3119863120598 we have
119881 (119860119898) ge 119881(
1003816100381610038161003816100381610038161003816120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897)
minus 119909(119894 minus 1
119897) minus 119909 (
119894 minus 2
119897)1003816100381610038161003816100381610038161003816le
120598
3119897 2 le 119894 le 119897)
(22)
Let 119899119898119897 be a positive integer for any 119898 ge 1 then by the
definition of 120578119899119898
(see (10)) it follows that for 2 le 119894 le 119897 119897 ge3119863120598 and119898 ge 1
120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897) =
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898
(23)
In addition since 119909 isin 119869(120583 120583) we know that
119886119894
119897≜ 119909 (
119894 minus 1
119897) minus 119909 (
119894 minus 2
119897) isin [
120583
119897120583
119897]
forall2 le 119894 le 119897 119897 ge3119863
120598
(24)
Then it follows that
119881 (119860119898) ge 119881(
119897
⋂119894=2
100381610038161003816100381610038161003816100381610038161003816
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894
100381610038161003816100381610038161003816100381610038161003816le120598
3)
forall119898 ge 1
(25)
For 2 le 119894 le 119897 and 120575 isin (0 1205983) we set
119892120575
(119910) =
1 119910 isin [minus120598
3+ 120575
120598
3minus 120575]
119910 + (1205983)
120575 119910 isin (minus
120598
3 minus
120598
3+ 120575)
(1205983) minus 120575 minus 119910
120575 119910 isin (
120598
3minus 120575
120598
3)
0 119910 isin [120598
3 +infin) cup (minusinfin minus
120598
3]
(26)
Obviously Π119897119894=2119892120575
(119910119894) is a continuous function on R119897minus1
satisfying linear growth condition Since 119883119899infin
119899=1is indepen-
dent under E (see Definition 2) from (25) we have
119881 (119860119898) ge E[
119897
prod119894=2
119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
=
119897
prod119894=2
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
(27)
Abstract and Applied Analysis 5
Since for all 2 le 119894 le 119897 with 119897 ge 3119863120598 and 119899 ge 1 E[119883119899minus
119886119894] = 120583minus119886
119894E[119883
119899minus119886119894] = 120583minus119886
119894and sup
119899ge1E[|119883119899minus119886119894|2
] lt infinlet 119899119898tend toinfin as119898 tends toinfin then by Lemma 6 we have
lim119898rarrinfin
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
= sup120583minus119886119894le119906le120583minus119886
119894
119892120575
(119906) = 1
(28)
since 119886119894isin [120583 120583] for 2 le 119894 le 119897 Thus from (27) and (28) it fol-
lows that lim inf119898rarrinfin
119881(119860119898) ge 1 Obviously 119881(119860
119898) le 1 for
all119898 ge 1 Hence this theorem follows
Corollary 8 Let 120593 be a real-valued continuous functional on119862[0 1] then for any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists a sub-sequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038161003816100381610038161003816120593 (120578119899119898
) minus 120593 (119909)10038161003816100381610038161003816le 120598) = 1 (29)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598In particular if we assume that 120593(119909) = 119909(1) for all 119909 isin
119862[0 1] then we have
lim119898rarrinfin
119881(
100381610038161003816100381610038161003816100381610038161003816
119878119899119898
119899119898
minus 119909 (1)
100381610038161003816100381610038161003816100381610038161003816le 120598) = 1 (30)
where 119909(1) isin [120583 120583]
5 Strong Limit Theorem underContinuous Upper Probability
In the previous Sections 2ndash4 we consider the general upperprobability 119881 For the sake of technique in this section wefurther assume that 119881 is continuous and investigate a stronglimit theorem of 120578
119899infin
119899=1under such a continuous upper
probability 119881 and its extension
51 Strong Limit Theorem
Theorem 9 Any 119909 isin 119869(120583 120583) is a limit point of somesubsequence of 120578
119899infin
119899=1wp 1 under 119881 that is
119881 (119909 isin Clust 120578119899) = 1 (31)
where Clust119909119899 denotes the cluster set of all the limit points of
real sequence 119909119899infin
119899=1
Proof From Lemma 5 since 119881 is continuous we only needto prove that for any 119909 isin 119869(120583 120583) and any 120598 gt 0
119881(lim inf119899rarrinfin
1003817100381710038171003817120578119899 minus 1199091003817100381710038171003817 le 120598)
= 119881(
infin
⋂119899=1
infin
⋃119898=119899
1003817100381710038171003817120578119898 minus 119909
1003817100381710038171003817 le 120598) = 1
(32)
Let 119860119898infin
119898=1and 119863 be defined the same as in the proof
of Theorem 7 Then it is sufficient to prove that for any fixed120598 gt 0 we can find a subsequence 119899
119898infin
119898=1of N such that
119881(lim inf119898rarrinfin
120578119899119898
minus 119909 le 120598) = 119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = 1 (33)
Take 119899119898= 119897119898 for 119898 ge 1 where 119897 ge 3119863120598 is an integer
FromTheorem 7 and the continuity of 119881 we can get
119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = lim119898rarrinfin
119881(
infin
⋃119895=119898
119860119895) ge lim119898rarrinfin
119881 (119860119898) = 1
(34)
Thus this theorem is proved
Remark 10 From the proof of Theorem 9 we can see thatit is implied by weak limit Theorem 7 under continuousupper probability It seems that ldquoweak limit theoremrdquo isstronger than ldquostrong limit theoremrdquo under continuous upperprobability IfP is a singleton thus we have 120583 = 120583 Then ourldquostrong limit theoremrdquo is not the same form as the strong lawof large numbers for sequences of random variables since theformer form is related to inferior limit and the latter one isrelated to limit
52 Extension to Functional Random Variables ByTheorem 9 we can easily get the following limit resultfor functional random variables
Corollary 11 Let 120593 be a real-valued continuous functionaldefined on 119862[0 1] then we have for any 119909 isin 119869(120583 120583)
119881 (120593 (119909) isin Clust 120593 (120578119899)) = 1 (35)
In particular
119881( sup119909isin119869(120583120583)
120593 (119909) le lim sup119899rarrinfin
120593 (120578119899))
= 119881( inf119909isin119869(120583120583)
120593 (119909) ge lim inf119899rarrinfin
120593 (120578119899)) = 1
(36)
From the proof ofTheorem 31 and Corollary 32 of Chenet al [11] the following lemma can be easily obtained
Lemma 12 Supposing 119891 is a real-valued continuous functionon R then
V( inf119910isin[120583120583]
119891 (119909) le lim inf119899rarrinfin
119891(119878119899
119899)
le lim sup119899rarrinfin
119891(119878119899
119899) le sup119910isin[120583120583]
119891 (119909)) = 1
(37)
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Lemma 6 Given a sequence of independent random variables119884119899infin
119899=1underE we assume that there exist two constants 119886 lt 119887
such that E[119884119899] = 119886 and E[119884
119899] = 119887 for all 119899 ge 1 and we
also assume that sup119899ge1
E[|119884119899|2
] lt infin Then for any increasingsubsequence 119899
119896infin
119896=1of N satisfying 119899
119896minus 119899119896minus1
converges to infinas 119899 tends toinfin and for any 120593 isin 119862(119877) with linear growth wehave
lim119896rarrinfin
E [120593(119878119899119896
minus 119878119899119896minus1
119899119896minus 119899119896minus1
)] = sup119886le119906le119887
120593 (119906) (18)
where 119878119898= sum119898
119894=1119884119894for all119898 ge 1
4 Weak Limit Theorem
In this section we will investigate the weak convergenceproblem of 120578
119899infin
119899=1under general upper probability
Theorem 7 For any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists asubsequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598) = 1 (19)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598
Proof For any 119909 isin 119869(120583 120583) and 120598 gt 0 by Lemma 5 we onlyneed to find a subsequence 120578
119899119898
infin
119898=1satisfying (19) Set
119860119898= 120596 isin Ω
10038171003817100381710038171003817120578119899119898
minus 11990910038171003817100381710038171003817le 120598 forall119898 ge 1 (20)
Note that for any integer 119897 ge 1
119881 (119860119898)
= 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)
+ 119909(119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
ge 119881( sup119905isin[(119894minus1)119897 119894119897] 1le119894le119897
1003816100381610038161003816100381610038161003816120578119899119898(119905) minus 120578
119899119898
(119894 minus 1
119897)
+ 120578119899119898
(119894 minus 1
119897) minus 119909 (
119894 minus 1
119897)1003816100381610038161003816100381610038161003816
+ sup119905isin[(119894minus1)119897 119894119897]1le119894le119897
1003816100381610038161003816100381610038161003816119909 (
119894 minus 1
119897) minus 119909 (119905)
1003816100381610038161003816100381610038161003816le 120598)
(21)
Denoting119863 = max|120583| |120583| since 119909 isin 119869(120583 120583) thus for all1 le 119894 le 119897 |119909((119894 minus 1)119897) minus 119909(119905)| le 119863119897 for all 119905 isin [(119894 minus 1)119897 119894119897]Hence taking 119897 ge 3119863120598 we have
119881 (119860119898) ge 119881(
1003816100381610038161003816100381610038161003816120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897)
minus 119909(119894 minus 1
119897) minus 119909 (
119894 minus 2
119897)1003816100381610038161003816100381610038161003816le
120598
3119897 2 le 119894 le 119897)
(22)
Let 119899119898119897 be a positive integer for any 119898 ge 1 then by the
definition of 120578119899119898
(see (10)) it follows that for 2 le 119894 le 119897 119897 ge3119863120598 and119898 ge 1
120578119899119898
(119894 minus 1
119897) minus 120578119899119898
(119894 minus 2
119897) =
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898
(23)
In addition since 119909 isin 119869(120583 120583) we know that
119886119894
119897≜ 119909 (
119894 minus 1
119897) minus 119909 (
119894 minus 2
119897) isin [
120583
119897120583
119897]
forall2 le 119894 le 119897 119897 ge3119863
120598
(24)
Then it follows that
119881 (119860119898) ge 119881(
119897
⋂119894=2
100381610038161003816100381610038161003816100381610038161003816
119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894
100381610038161003816100381610038161003816100381610038161003816le120598
3)
forall119898 ge 1
(25)
For 2 le 119894 le 119897 and 120575 isin (0 1205983) we set
119892120575
(119910) =
1 119910 isin [minus120598
3+ 120575
120598
3minus 120575]
119910 + (1205983)
120575 119910 isin (minus
120598
3 minus
120598
3+ 120575)
(1205983) minus 120575 minus 119910
120575 119910 isin (
120598
3minus 120575
120598
3)
0 119910 isin [120598
3 +infin) cup (minusinfin minus
120598
3]
(26)
Obviously Π119897119894=2119892120575
(119910119894) is a continuous function on R119897minus1
satisfying linear growth condition Since 119883119899infin
119899=1is indepen-
dent under E (see Definition 2) from (25) we have
119881 (119860119898) ge E[
119897
prod119894=2
119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
=
119897
prod119894=2
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
(27)
Abstract and Applied Analysis 5
Since for all 2 le 119894 le 119897 with 119897 ge 3119863120598 and 119899 ge 1 E[119883119899minus
119886119894] = 120583minus119886
119894E[119883
119899minus119886119894] = 120583minus119886
119894and sup
119899ge1E[|119883119899minus119886119894|2
] lt infinlet 119899119898tend toinfin as119898 tends toinfin then by Lemma 6 we have
lim119898rarrinfin
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
= sup120583minus119886119894le119906le120583minus119886
119894
119892120575
(119906) = 1
(28)
since 119886119894isin [120583 120583] for 2 le 119894 le 119897 Thus from (27) and (28) it fol-
lows that lim inf119898rarrinfin
119881(119860119898) ge 1 Obviously 119881(119860
119898) le 1 for
all119898 ge 1 Hence this theorem follows
Corollary 8 Let 120593 be a real-valued continuous functional on119862[0 1] then for any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists a sub-sequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038161003816100381610038161003816120593 (120578119899119898
) minus 120593 (119909)10038161003816100381610038161003816le 120598) = 1 (29)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598In particular if we assume that 120593(119909) = 119909(1) for all 119909 isin
119862[0 1] then we have
lim119898rarrinfin
119881(
100381610038161003816100381610038161003816100381610038161003816
119878119899119898
119899119898
minus 119909 (1)
100381610038161003816100381610038161003816100381610038161003816le 120598) = 1 (30)
where 119909(1) isin [120583 120583]
5 Strong Limit Theorem underContinuous Upper Probability
In the previous Sections 2ndash4 we consider the general upperprobability 119881 For the sake of technique in this section wefurther assume that 119881 is continuous and investigate a stronglimit theorem of 120578
119899infin
119899=1under such a continuous upper
probability 119881 and its extension
51 Strong Limit Theorem
Theorem 9 Any 119909 isin 119869(120583 120583) is a limit point of somesubsequence of 120578
119899infin
119899=1wp 1 under 119881 that is
119881 (119909 isin Clust 120578119899) = 1 (31)
where Clust119909119899 denotes the cluster set of all the limit points of
real sequence 119909119899infin
119899=1
Proof From Lemma 5 since 119881 is continuous we only needto prove that for any 119909 isin 119869(120583 120583) and any 120598 gt 0
119881(lim inf119899rarrinfin
1003817100381710038171003817120578119899 minus 1199091003817100381710038171003817 le 120598)
= 119881(
infin
⋂119899=1
infin
⋃119898=119899
1003817100381710038171003817120578119898 minus 119909
1003817100381710038171003817 le 120598) = 1
(32)
Let 119860119898infin
119898=1and 119863 be defined the same as in the proof
of Theorem 7 Then it is sufficient to prove that for any fixed120598 gt 0 we can find a subsequence 119899
119898infin
119898=1of N such that
119881(lim inf119898rarrinfin
120578119899119898
minus 119909 le 120598) = 119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = 1 (33)
Take 119899119898= 119897119898 for 119898 ge 1 where 119897 ge 3119863120598 is an integer
FromTheorem 7 and the continuity of 119881 we can get
119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = lim119898rarrinfin
119881(
infin
⋃119895=119898
119860119895) ge lim119898rarrinfin
119881 (119860119898) = 1
(34)
Thus this theorem is proved
Remark 10 From the proof of Theorem 9 we can see thatit is implied by weak limit Theorem 7 under continuousupper probability It seems that ldquoweak limit theoremrdquo isstronger than ldquostrong limit theoremrdquo under continuous upperprobability IfP is a singleton thus we have 120583 = 120583 Then ourldquostrong limit theoremrdquo is not the same form as the strong lawof large numbers for sequences of random variables since theformer form is related to inferior limit and the latter one isrelated to limit
52 Extension to Functional Random Variables ByTheorem 9 we can easily get the following limit resultfor functional random variables
Corollary 11 Let 120593 be a real-valued continuous functionaldefined on 119862[0 1] then we have for any 119909 isin 119869(120583 120583)
119881 (120593 (119909) isin Clust 120593 (120578119899)) = 1 (35)
In particular
119881( sup119909isin119869(120583120583)
120593 (119909) le lim sup119899rarrinfin
120593 (120578119899))
= 119881( inf119909isin119869(120583120583)
120593 (119909) ge lim inf119899rarrinfin
120593 (120578119899)) = 1
(36)
From the proof ofTheorem 31 and Corollary 32 of Chenet al [11] the following lemma can be easily obtained
Lemma 12 Supposing 119891 is a real-valued continuous functionon R then
V( inf119910isin[120583120583]
119891 (119909) le lim inf119899rarrinfin
119891(119878119899
119899)
le lim sup119899rarrinfin
119891(119878119899
119899) le sup119910isin[120583120583]
119891 (119909)) = 1
(37)
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Since for all 2 le 119894 le 119897 with 119897 ge 3119863120598 and 119899 ge 1 E[119883119899minus
119886119894] = 120583minus119886
119894E[119883
119899minus119886119894] = 120583minus119886
119894and sup
119899ge1E[|119883119899minus119886119894|2
] lt infinlet 119899119898tend toinfin as119898 tends toinfin then by Lemma 6 we have
lim119898rarrinfin
E [119892120575
(119878(119894minus1)119899
119898119897minus 119878(119894minus2)119899
119898119897
119899119898119897
minus 119886119894)]
= sup120583minus119886119894le119906le120583minus119886
119894
119892120575
(119906) = 1
(28)
since 119886119894isin [120583 120583] for 2 le 119894 le 119897 Thus from (27) and (28) it fol-
lows that lim inf119898rarrinfin
119881(119860119898) ge 1 Obviously 119881(119860
119898) le 1 for
all119898 ge 1 Hence this theorem follows
Corollary 8 Let 120593 be a real-valued continuous functional on119862[0 1] then for any 119909 isin 119869(120583 120583) and 120598 gt 0 there exists a sub-sequence 120578
119899119898
infin
119898=1such that
lim119898rarrinfin
119881(10038161003816100381610038161003816120593 (120578119899119898
) minus 120593 (119909)10038161003816100381610038161003816le 120598) = 1 (29)
where 119899119898infin
119898=1is an increasing subsequence of N and depends
on 120583 120583 and 120598In particular if we assume that 120593(119909) = 119909(1) for all 119909 isin
119862[0 1] then we have
lim119898rarrinfin
119881(
100381610038161003816100381610038161003816100381610038161003816
119878119899119898
119899119898
minus 119909 (1)
100381610038161003816100381610038161003816100381610038161003816le 120598) = 1 (30)
where 119909(1) isin [120583 120583]
5 Strong Limit Theorem underContinuous Upper Probability
In the previous Sections 2ndash4 we consider the general upperprobability 119881 For the sake of technique in this section wefurther assume that 119881 is continuous and investigate a stronglimit theorem of 120578
119899infin
119899=1under such a continuous upper
probability 119881 and its extension
51 Strong Limit Theorem
Theorem 9 Any 119909 isin 119869(120583 120583) is a limit point of somesubsequence of 120578
119899infin
119899=1wp 1 under 119881 that is
119881 (119909 isin Clust 120578119899) = 1 (31)
where Clust119909119899 denotes the cluster set of all the limit points of
real sequence 119909119899infin
119899=1
Proof From Lemma 5 since 119881 is continuous we only needto prove that for any 119909 isin 119869(120583 120583) and any 120598 gt 0
119881(lim inf119899rarrinfin
1003817100381710038171003817120578119899 minus 1199091003817100381710038171003817 le 120598)
= 119881(
infin
⋂119899=1
infin
⋃119898=119899
1003817100381710038171003817120578119898 minus 119909
1003817100381710038171003817 le 120598) = 1
(32)
Let 119860119898infin
119898=1and 119863 be defined the same as in the proof
of Theorem 7 Then it is sufficient to prove that for any fixed120598 gt 0 we can find a subsequence 119899
119898infin
119898=1of N such that
119881(lim inf119898rarrinfin
120578119899119898
minus 119909 le 120598) = 119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = 1 (33)
Take 119899119898= 119897119898 for 119898 ge 1 where 119897 ge 3119863120598 is an integer
FromTheorem 7 and the continuity of 119881 we can get
119881(
infin
⋂119898=1
infin
⋃119895=119898
119860119895) = lim119898rarrinfin
119881(
infin
⋃119895=119898
119860119895) ge lim119898rarrinfin
119881 (119860119898) = 1
(34)
Thus this theorem is proved
Remark 10 From the proof of Theorem 9 we can see thatit is implied by weak limit Theorem 7 under continuousupper probability It seems that ldquoweak limit theoremrdquo isstronger than ldquostrong limit theoremrdquo under continuous upperprobability IfP is a singleton thus we have 120583 = 120583 Then ourldquostrong limit theoremrdquo is not the same form as the strong lawof large numbers for sequences of random variables since theformer form is related to inferior limit and the latter one isrelated to limit
52 Extension to Functional Random Variables ByTheorem 9 we can easily get the following limit resultfor functional random variables
Corollary 11 Let 120593 be a real-valued continuous functionaldefined on 119862[0 1] then we have for any 119909 isin 119869(120583 120583)
119881 (120593 (119909) isin Clust 120593 (120578119899)) = 1 (35)
In particular
119881( sup119909isin119869(120583120583)
120593 (119909) le lim sup119899rarrinfin
120593 (120578119899))
= 119881( inf119909isin119869(120583120583)
120593 (119909) ge lim inf119899rarrinfin
120593 (120578119899)) = 1
(36)
From the proof ofTheorem 31 and Corollary 32 of Chenet al [11] the following lemma can be easily obtained
Lemma 12 Supposing 119891 is a real-valued continuous functionon R then
V( inf119910isin[120583120583]
119891 (119909) le lim inf119899rarrinfin
119891(119878119899
119899)
le lim sup119899rarrinfin
119891(119878119899
119899) le sup119910isin[120583120583]
119891 (119909)) = 1
(37)
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
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
6 Abstract and Applied Analysis
Corollary 13 Let 119891 be defined the same as Lemma 12 then
119881(lim sup119899rarrinfin
119891(119878119899
119899) = sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) = inf119910isin[120583120583]
119891 (119909)) = 1
(38)
Especially if we assume 119891(119909) = 119909 for all 119909 isin R then
119881(lim sup119899rarrinfin
119878119899
119899= 120583) = 119881(lim inf
119899rarrinfin
119878119899
119899= 120583) = 1 (39)
Proof Take 120593(119909) = 119891(119909(1)) forall119909 isin 119862[0 1] It is easy to checkthat 120593 is a continuous functional on 119862[0 1] and obviously120593(119909) isin [120583 120583] For any 119899 ge 1 120593(120578
119899) = 119891(120578
119899(1)) = 119891(119878
119899119899)
Thus from Corollary 11 it follows that
119881(lim sup119899rarrinfin
119891 (119878119899
119899) ge sup119910isin[120583120583]
119891 (119909))
= 119881(lim inf119899rarrinfin
119891(119878119899
119899) le inf119910isin[120583120583]
119891 (119909)) = 1
(40)
Then this corollary follows from (37) of Lemma 12 and(40)
53 Inequalities In this subsection we will give some usefulexamples as applications in inequalities
Example 14 Let 119891 be a Lebesgue integrable function definedfrom [0 1] to R we denote 119865(119905) = int
1
119905
119891(119904)119889119904 119905 isin [0 1] Then
lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992le int1
0
119865 (119905) 1198921(119865 (119905)) 119889119905 (41)
lim sup119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992ge int1
0
119865 (119905) 1198922(119865 (119905)) 119889119905 (42)
hold wp 1 under 119881 respectively where
1198921(119910) =
120583 119910 ge 0
120583 119910 lt 01198922(119910) =
120583 119910 ge 0
120583 119910 lt 0(43)
Especially for 119891 equiv 1 we have wp 1 under 119881 respectively
lim inf119899rarrinfin
119899
sum119894=1
119878119894
1198992le120583
2 lim sup
119899rarrinfin
119899
sum119894=1
119878119894
1198992ge120583
2 (44)
Proof Observe that 120593(119909) = int1
0
119891(119905)119909(119905)119889119905 for all 119909 isin 119862[0 1]
is a continuous functional defined from 119862[0 1] to R And itis easy to check that wp 1 under V
lim inf119899rarrinfin
120593 (120578119899) = lim inf119899rarrinfin
int1
0
119891 (119905) 120578119899(119905) 119889119905
= lim inf119899rarrinfin
119899
sum119894=1
119891(119894
119899)119878119894
1198992
(45)
By Corollary 11 we know that wp 1 under 119881
lim inf119899rarrinfin
120593 (120578119899) le inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 (46)
Since for any 119909 isin 119869(120583 120583) 1199091015840(119905) isin [120583 120583] almosteverywhere for 119905 isin [0 1] then note that for all 119909 isin 119869(120583 120583)
inf119909isin119869(120583120583)
int1
0
119891 (119905) 119909 (119905) 119889119905 = inf119909isin119869(120583120583)
int1
0
119865 (119905) 1199091015840
(119905) 119889119905
le int1
0
119865 (119905) 1198921(119905) 119889119905
(47)
Thus inequality (41) holds wp 1 under 119881 The proof ofinequality (42) is similar to inequality (41) and inequalities(44) are obvious We complete the whole proof
Example 15 For any integer 119896 ge 1 we have that
lim inf119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1le
min 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
lim sup119899rarrinfin
10038161003816100381610038161198781198941003816100381610038161003816119896
119896 + 1ge
max 1003816100381610038161003816100381612058310038161003816100381610038161003816
119896
10038161003816100381610038161205831003816100381610038161003816119896
119896 + 1
(48)
hold wp 1 under 119881 respectively
Proof It is easy to check that 120593(119909) = int1
0
|119909(119905)|119896
119889119905 is acontinuous functional on 119862[0 1] Thus this example can besimilarly proved as Example 14
6 Applications in Finance
We consider a capital market with ambiguity which ischaracterized by a set of probabilities denoted the same asprevious sections by P such that the corresponding upperprobability119881 is continuous For simplicity let risk free rate bezero We will investigate the stock price 119878
119905over time interval
[0 1] on themeasurable space (ΩF) andwe assume that theincrements Δ119878
119905= 119878119905+Δ119905
minus 119878119905of stock price 119878
119905in time period
[119905 119905 + Δ119905] is independent from 119878119905for all 119905 119905 + Δ119905 isin [0 1]
that is for each probability 119876 isin P Δ119878119905and 119878
119905are mutually
independent under119876 for all 119905 119905+Δ119905 isin [0 1] We also assumethat the price of the stock is uniformly bounded with respectto (119905 120596) isin [0 1] times Ω and the largest and smallest expectedaverage return of this stock over time interval [119905 119905 + Δ119905] are120583 and 120583 respectively that is
E[Δ119878119905
Δ119905] = sup119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
minusE[minusΔ119878119905
Δ119905] = inf119876isinP
119864119876[Δ119878119905
Δ119905] = 120583
(49)
where minusinfin lt 120583 le 120583 lt infin and 119905 119905 + Δ119905 isin [0 1]
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
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
Abstract and Applied Analysis 7
For any 119899 ge 1 take Δ119905 = 1119899 and let 119883119896= (Δ119878
(119896minus1)119899)
(1119899) and 119878119896= sum119896
119894=1119883119894for 1 le 119896 le 119899 Then it is obvious that
119883119896119899
119896=1is a sequence of independent random variables inM
under upper probability E with supermean E[119883119896] = 120583 and
submean minusE[minus119883119896] = 120583 for all 1 le 119896 le 119899 Denote the average
stock price of 119878119896119899119899
119896=1by int10
119878119899
119905119889119905 then
int1
0
119878119899
119905(120596) 119889119905 =
119899
sum119896=1
119878119896119899
(120596)
119899=
119899
sum119896=1
sum119896
119894=1Δ119878(119894minus1)119899
(120596)
119899
=
119899
sum119896=1
sum119896
119894=1119883119894(120596)
1198992=
119899
sum119896=1
119878119896(120596)
1198992 forall120596 isin Ω
(50)
Then by inequalities (41) and (42) it follows that
lim inf119899rarrinfin
int1
0
119878119899
119905119889119905 le
120583
2 lim sup
119899rarrinfin
int1
0
119878119899
119905119889119905 ge
120583
2(51)
hold respectively wp 1 under continuous upper probability119881 (Together with X Chen and Z Chen [15] we will see thatthese two inequalities can become equalities in the future)
7 Concluding Remarks
This paper proves that any element of subset 119869(120583 120583) ofcontinuous function space on [0 1] is a limit point of certainsubsequence of stochastic processes 120578
119899in upper probability
119881 and with probability 1 under continuous upper probabilityIt is an extension of strong law of large numbers fromrandom variables to stochastic processes in the framework ofupper probability The limit theorem for functional randomvariables also is proved It is very useful in finance whenthere is ambiguity But the constraint conditions in this paperare very strong such as the condition E[sup
119899ge11198832
119899] lt infin
and independence under sublinear expectation How canwe weaken the constraint conditions Does the strong limittheoremunder upper probability still holdwithout continuityof 119881We will investigate them in the future work
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was partially supported by the WCU (WorldClass University) Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (R31-20007) and was also partiallysupported by the National Natural Science Foundation ofChina (no 11231005)
References
[1] G Choquet ldquoTheory of capacitiesrdquoAnnales de Linstitut Fouriervol 5 no 131ndash295 p 87 1953
[2] N El Karoui S Peng and M C Quenez ldquoBackward stochasticdifferential equations in financerdquo Mathematical Finance vol 7no 1 pp 1ndash71 1997
[3] S Peng ldquoNonlinear expectations and stochastic calculus underuncertaintyrdquo 2010 httparxivorgabs10024546
[4] S Peng ldquoLaw of large numbers and central limit theoremunder nonlinear expectationsrdquo 2007 httparxivorgpdfmath0702358pdf
[5] S Peng ldquoA new central limit theorem under sublinear expecta-tionsrdquo 2008 httparxivorgpdf08032656pdf
[6] Z Chen and F Hu ldquoA law of the iterated logarithm for sublinearexpectationsrdquo 2013 httparxivorgpdf11032965pdf
[7] PWuandZChen ldquoAn invariance principle ofG-Brownianmo-tion for the law of the iterated logarithm under G-expectationrdquoMay 2011 httparxivorgabs11050135
[8] M Marinacci ldquoLimit laws for non-additive probabilities andtheir frequentist interpretationrdquo Journal of Economic Theoryvol 84 no 2 pp 145ndash195 1999
[9] P Teran ldquoLaws of large numbers without additivityrdquo Transac-tions of American Mathematical Society Accepted for publica-tion
[10] Z Chen ldquoStrong laws of large numbers for capacitiesrdquo 2010httparxivorgabs10060749
[11] Z Chen P Wu and B Li ldquoA strong law of large numbersfor non-additive probabilitiesrdquo International Journal of Approx-imate Reasoning vol 54 no 3 pp 365ndash377 2013
[12] J Y Halpern Reasoning about Uncertainty MIT Press Cam-bridge Mass USA 2003
[13] X Chen ldquoStrong law of large numbers under an upper proba-bilityrdquo Applied Mathematics vol 3 no 12 pp 2056ndash2062 2013
[14] X Chen ldquoAn invariance principle of SLLN for G-quadraticvariational process under capacitiesrdquo in Proceedings of the Inter-national Conference on Information Technology and ComputerApplication Engineering (ITCAE rsquo13) pp 113ndash117 CRC Press2013
[15] X Chen and Z Chen ldquoAn invariance principle of strong law oflarge numbers under non-additive probabilitiesrdquo Preprint
[16] S Peng ldquoSurvey on normal distributions central limit theoremBrownian motion and the related stochastic calculus undersublinear expectationsrdquo Science in China A vol 52 no 7 pp1391ndash1411 2009
[17] F Hu ldquoGeneral laws of large numbers under sublinear expecta-tionsrdquo 2011 httparxivorgabs11045296
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Function Spaces
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International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
![Annealed and Quenched Limit Theorems for Random Expanding ... › download › pdf › 52434616.pdf · dent probabilities, see [8] and references therein. For limit theorems, the](https://static.fdocuments.in/doc/165x107/5f02e26c7e708231d4067d73/annealed-and-quenched-limit-theorems-for-random-expanding-a-download-a-pdf.jpg)
