In this paper, variable exponent function spaces Lp·, Lbp·, and Lcp· are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.

Natural Science Foundation of Fujian Province2016J050081. Introduction

Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering. Basic properties of the spaces are first given by Kováčik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2, 3]. Harjulehto et al. present an overview of applications to differential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of variable exponent function spaces and applications to partial differential equations. In [6], Aoyama proves some important probability inequalities in variable exponent Lebesgue spaces.

Nonlinear expectations play an important role in the research of financial markets. One of the most important application is that a coherent risk measure(the basic theory about coherent risk measure can be found in [7]) is a sublinear expectation E:H→R defined on H, which is a linear space of finacial losses. In this paper, we are interested in behavior of sublinear expectation spaces with variable exponents. By the following representation theorem which can be found in Peng ([8], p. 4), we know that a sublinear expectation can be expressed as a supremum of linear expectations, i.e., there exists a family of linear expectations {Eθ}θ∈Θ such that(1)EX=supθ∈ΘEθX,X∈H.

Thus, we consider the upper expectation only in this paper. Some other important theories about nonlinear expectations can be found in Peng’s [9, 10].

The remainder of the paper is divided as follows: in Section 2, motivated by Fu [11] and Denis et al. [12], we first introduce Lp·, Lbp· and Lcp· and give some properties of these spaces. And Each element of ·p·-completion of CbΩ has a quasi-continuous version is proved. In Section 3, applying the results of Section 2, we discuss a version of Kolmogorov’s criterion for continuous modification of stochastic processes, which are in the variable exponent function space related to a sublinear expectation, after proving the situation under a linear expectation.

2. Variable Exponent Function Spaces

Let Ω be a complete metric space equipped with the distance d, BΩ the Borel σ− algebra of Ω and M the collection of all probability measures on Ω,BΩ.

L0Ω: the space of all BΩ-measurable real functions;

BbΩ: all bounded functions in L0Ω;

CbΩ: all continuous functions in BbΩ.

For a given subset P⊆M, we denote(2)cA≔supP∈PPA,A∈BΩ.

Definition 1.

A set A is a polar if cA=0 and a property holds “quasi-surely” (q.s.) if it holds outside a polar set.

The upper expectation of P is defined as follows: for each X∈L0Ω such that EPX exists for each P∈P,(3)EX≔supP∈PEPX.

About E, we know the following properties.

Theorem 2 (see Theorem 9 in [<xref ref-type="bibr" rid="B12">12</xref>]).

The upper expectation E· of family P is a sublinear expectation on BbΩ as well as on CbΩ, i.e.,

For all X, YinBbΩ, X≥Y, EX≥EY.

For all X, YinBbΩ, EX+Y≤EX+EY.

For all λ≥0, X∈BbΩ, EλX=λEX.

For all c∈R, X∈BbΩ, EX+c=EX+c.

Let a BΩ-measurable real function p:Ω→1,∞, be a variable exponent. In the space L0Ω, the moduli ρ are defined by(4)ρp·X≔EXp=supP∈EPXp=supP∈∫ΩXωpωPdω.

Definition 3.

The space Lp· is the set of X∈L0Ω satisfying ρp·X<∞, and it is endowed with the Luxemburg norm:

(5)Xp·=infλ>0:ρp·Xλ≤1.

We set Np·:=X∈L0Ω:ρX=0, and denote Lp·:=Lp·/Np·. As usual, we do not take care about the distinction between classes and their representatives.

Proposition 4.

If the variable exponent p satisfies 1<p−≤pω≤p+<∞, then the inequality(6)EXY≤1+1p−−1p+Xp·Yp′·

holds for every X∈Lp· and Y∈Lp′·, where 1/pω+1/p′ω=1.

Proof.

By Young inequality, we have

(7)XXp·YYp′·≤1pωXXp·p+1p′ωYYp′·p′.

By the monotonicity, sub-additivity and positive homogeneity of E, and the property of the norm, we have(8)EXXp·YYp′·≤E1pωXXp·p+1p′ωYYp′·p′≤E1pωXXp·p+E1p′ωYYp′·p′≤1p−EXXp·p+1−1p+EYYp′·p′≤1+1p−−1p+.

Thus, the inequality follows.

Proposition 5.

Suppose that the variable exponent p satisfies 1<p−≤pω≤p+<∞. If X,Xk∈Lp·, then

If Xp·≥1, then Xp·p−≤ρp·X≤Xp·p+.

If Xp·≤1, then Xp·p+≤ρp·X≤Xp·p−.

limk→∞Xkp·=0, if and only if limk→∞ρp·Xk=0

limk→∞Xkp·=∞, if and only if limk→∞ρp·Xk=∞.

In particular, the linear expectation EP also follows this proposition.

Proof.

(1) By XPp·≥1 and the definition of the norm,

(9)EXpXp·p+≤EXXp·p≤1.

Thus, ρp·X≤Xp·p+. As(10)Xp·p−/p≤Xp·,

we also have(11)EXXp·p−/pp≥1.

That is to say Xp·p−≤ρp·X. The proof is completed. (2) can be easily proved in the similar method. By (1) and (2), we can easily get (3) and (4).

Proposition 6.

Suppose that the variable exponent p satisfies 1<p−≤pω≤p+<∞. Let X,Xk∈Lp·. Then for each α>0(12)cX>α≤α−p−∨α−p+EXp.

Proof.

For each P∈P, by Markov inequality, we have

(13)PXαp>1≤EPXαp.

Take the supremum in P both sides,(14)cXαp>1≤EXαp

Thus,(15)cX>α≤EXαp≤α−p−∨α−p+EXp.

Lemma 7 (Proposition 17 in [<xref ref-type="bibr" rid="B12">12</xref>]).

Let p∈0,∞ and {Xn} be a sequence in Lp which converges to X in Lp. Then there exists a subsequence {Xnk} which converges to X quasi-surely in the sense that it converges to X outside a polar set.

Proposition 8.

If the variable exponent p satisfies 1<p−≤pω≤p+<∞. Lp· is a Banach space.

Proof.

Let {Xn} be a Cauchy sequence in Lp·. Then, by Proposition 4,(16)EXm−Xn≤CXm−Xnp·,

where C is a constant. That is to say {Xn} is a Cauchy sequence in L1. By Proposition 14 in [12], L1 is a Banach Space. Thus, {Xn} converges in L1. Suppose that Xn→X, X∈L1 and further by Lemma 7, we suppose Xn→X quasi-surely (subtracting a subsequence if necessary). For each 0<ε<1, there exists n0 such that Xm−Xnp·<ε for m,n≥n0. Fix n, by Fatou’s lemma(17)EXn−Xεp=supP∈PEPXn−Xεp≤supP∈Pliminfm→∞EPXn−Xmεp≤liminfm→∞EXn−Xmεp≤1.

Thus, Xn−Xp·≤ε, and further ρp·Xn−X≤Xn−Xp·p−≤εp−, that is to say, Xn−X∈Lp·, and X∈Lp·. The proof is completed.

We denote by Lbp· the completion of BbΩ and by Lcp· the completion of CbΩ. By Proposition 8, we have Lcp·⊂Lbp·⊂Lp· for 1<p−≤pω≤p+<∞.

Proposition 9.

Suppose that the variable exponent p satisfies 1<p−≤pω≤p+<∞, then(18)Lbp·=X∈Lp·:limn→∞EXp1X>n=0.

Proof.

We denote Jp·={X∈Lp·:limn→∞EXp1{X>n}=0}. For each X∈Jp· let Xn=X∧n∨−n∈BbΩ. We have

(19)ρp·X−Xn≤EXp1X>n,

so limn→∞ρp·X−Xn=0. Thus, X∈Lbp·. On the other hand, for any X∈Lbp·, we can find a sequence {Yn}⊂BbΩ such that limn→∞ρp·X−Yn=0. Let yn=supω∈Ω|Ynω| and Xn=X∧yn∨−yn. Since |X−Xn≤X−Yn|, we have limn→∞ρp·X−Xn=0. This implies that for any sequence {αn} such that αn→∞ as n→∞, limn→∞ρp·X−X∧αn∨−αn=0. And for all n∈N,(20)EXp1X>n<2p+−1EX−np1X>n+E2p−1np1X>n.

For the first term of right hand side, we have(21)Limn→∞EX−np1X>n=limn→∞ρp·X−X∧n∨−n=0.

For the second term,(22)np2p1X>n≤X−n2p1X>n≤X−n2p1X>n/2,

If the variable exponent p satisfies 1<p−≤pω≤p+<∞, let X∈Lbp·, then for each ε>0, there exists a δ>0, such that for all A∈BΩ with cA≤δ, we have EXp1A≤ε.

Proof.

For each ε>0, by Proposition 9, there exists N>0 such that EXp1{X>n}≤ε/2. Set δ=ε/2Np+, then for A∈BΩ with cA≤δ, we have.

A mapping X on Ω with values in topological space is said to be quasi-continuous (q.c.) if ∀ε>0, there exists an open set O with cO<ε such that X|Oc is continuous.

Definition 12.

We say that X:Ω→R has a quasi-continuous version if there exists a quasi-continuous Y:Ω→R with X=Y q.s.

Proposition 13.

If the variable exponent p satisfies 1<p−≤pω≤p+<∞. Then each element in Lcp· has a quasi-continuous version.

Proof.

For each X∈Lcp·, there exists {Xn}⊂CbΩ such that Xn→X in Lcp·. Let us choose a subsequence {Xnk} such that

(25)ρp·Xnk+1−Xnk≤2−2k,∀k≥1,

and set for all k,(26)Ak=⋃i=k∞Xni+1−Xni>2−i/p+.

Because of the subadditivity property and Proposition 6,(27)cAk≤∑i=k∞cXni+1−Xni>2−i/p+≤∑i=k∞2iEXni+1−Xnip≤∑i=k∞2−i=2−k+1.

Thus, limk→∞cAk=0, so the Borel set A=⋂k=1∞Ak is polar. As each Xnk is continuous, for all k≥1, Ak is an open set. And for all k, {Xni} converges uniformly on Akc so that the limit is continuous on each Akc.

Proposition 14.

Suppose that the variable exponent p satisfies 1<p−≤pω≤p+<∞, then(28)Lcp·=X∈Lp·:Xhasaquasi‐continuousversion,andlimn→∞EXp1X>n=0.

For each X∈Lcp·, X has a quasi-continuous version by Proposition 13. Since X∈Lcp·⊂Lbp·, by Proposition 9 we have limn→∞EXp1{X>n}=0. Thus, X∈Jp·.

On the other hand, Let X∈Jp· be quasi-continuous. For all n∈N, define(30)Yn=X∧n∨−n.

Since limn→∞EXp1{X>n}=0 and(31)EX−Ynp≤EX−Ynp1X>n+EX−Ynp1X≤n≤EXp1X>n,

we have limn→∞E|X−Yn|p=0.

For all n∈N, Yn is quasi-continuous, so there exists a closed set Fn such that cFnc<1/np++1 and Yn is continuous on Fn. By Tietze’s extension theorem there exists Zn∈CbΩ such that |Zn|≤n and Zn=Yn on Fn. Then, we have (32)EYn−Znp≤EYn−Znp1Fn+EYn−Znp1Fnc≤E2np1Fnc≤2np+cFnc≤2p+n.

Suppose that X:Ω→R has a quasi-continuous version and that there exists a function f:R+→R+ satisfying limn→∞ft/tp=∞ uniformly on Ω and EfX<∞. Then X∈Lcp·, where 1<p−≤pω≤p+<∞.

Proof.

For each ε>0, there exists N>0 which is independent of ω such that for all t≥N,

(34)fttp≥1ε.

So(35)EXp1X>N≤εEfX1X>N≤εEfX.

Thus, limN→∞EXp1{X>N}=0.

3. Kolmogorov’s Criterion on Variable Exponent Function SpacesDefinition 16.

Let I be a set of indices. {Xt}t∈I and {Yt}t∈I be two processes indexed by I. We say that Y is a quasi-modification of X if for all t∈IXt=Yt q.s.

To prove a Kolmogorov’s criterion for a process indexed by Rd with d∈N on variable exponent function spaces, we give the following lemma first.

Lemma 17.

Let 1<p−≤pω≤p+<∞ and {Xt}t∈0,1d be a process such that for all t∈0,1d, Xt∈Lp·. Assume that for a fixed P∈P there exist positive constants c and ε such that

(36)EPXt−Xsp≤ct−sd+ε.

Then X admit a modification X˜ such that (37)EPsups≠tX∼t-X∼st-sαp<∞,

for every α∈0,ε/p+

Proof.

Fix P∈P. For m∈N, define the set of dyadic points in 0,1d:

(38)Dm≔i12m,⋯,id2m:i1⋯,id∈0,1,⋯,2m,

and D=∪mDm. Set Δm={s,t:|s−t|=2−m,s,t∈Dm}, and there are fewer than 2m+ld such pairs in Δm. For s,t∈D, we say that s≤t if each component of s is less than or equal to the corresponding component of t.

Set Ki=sups,t∈Δi|Xs−Xt|. And by the hypothesis in the lemma,(39)EPKip≤∑s,t∈ΔiEPXs,−,Xtp≤2i+1dc2−id+ε=c2d2−iε.

It is easy to see c2d2−iε<1 from some i0 on.

For s,t∈D, there exists increasing sequences {sn}⊂D and {tn}⊂D such that sn,tn∈Dn, sn≤s, tn≤t and sn=s, tn=t from some n on.

Let now s,t∈D and |s−t|≤2−m, and either sm=tm or sm,tm∈Δm and in any case,(40)Xs−Xt=∑i=m∞Xsi+1−Xsi+Xsm−Xtm+∑i=m∞Xti−Xti+1,

where the series are actually finite sums. It follows that(41)Xs−Xt≤Km+2∑i=m+1∞Ki≤2∑i=m∞Ki.

Thus, setting Mα=sups,t∈D,s≠tXs−Xt/s−tα, we have(42)Mα≤supm∈ℕ2m+1αsups,t∈D,s≠t,s−t≤2−mXs−Xt≤supm∈ℕ2m+1α+1∑i=m∞Ki≤2α+1∑i=0∞2iαKi.

for every α∈0,ε/p+, we have(44)Mαp·≤2α+1∑i=0∞2iαKip·≤2α+1∑i=0i0−12iαc2d2−iε1/p−+∑i=i0∞2iαc2d2−iε1/p+≤Cα,p−,ε+Cd,α,p+,ε∑i=i0∞2iα−ε/p+<∞,

where α∈0,ε/p+. It follows in particular that for almost every ω, X is uniformly continuous on D and it makes sense to set(45)X˜tω=lims→t,s∈DXsω.

By Fatou’s lemma and the hypothesis,(46)EPX˜t−Xtp≤liminfs→tEPXs−Xtp≤liminfs→tct−sd+ε,

so, X˜t=Xt a.s. and X˜t is a modification.

Theorem 18.

Let 1<p−≤pω≤p+<∞ and {Xt}t∈[0,1]d be a process such that for all t∈[0,1]d, Xt∈Lp·. Assume that there exists a positive constants c and ε such that(47)EXt−Xsp≤ct−sd+ε.

Then X admit a modification X˜ such that(48)Esups≠tX˜t−X˜st−sαp<∞,

for every α∈0,ε/p+.

Proof.

Let set D the same as the proof of Lemma 17. We set.

(49)Mα=sups,t∈D,s≠tXs−Xts−tα,

where α∈0,ε/p+. By Lemma 17, we know that for any P∈P, EPMp is finite and uniformly bounded with respect to P so that(50)EMp=supP∈PEPMp<∞.

Thus, X is uniformly continuous on D q.s. and set(51)X˜tω=lims→t,s∈DXsω,t∈0,1d.

In the similar way in Lemma 17, X˜t=Xt q.s. and X˜t is a modification.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The author gratefully acknowledges the support from Fujian Provincial Natural Science Foundation (2016J05008).

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