Random Field Theory Under Nonlinear Expectation And Related Topics

In 1933,Kolmogorov[55]first introduced the axiomatic formulation of probability space(?.F.P).Since est.ablished,it became an important mathematical framework for quantitative studies of random events.However,in most practical cases,the probability measure P is unknown.In 1921,the famous economist Knight[54]pointed out that in economics,probability and statistics models have intrinsic and inevitable uncertainties.This probability uncertainty is also called Knightian uncertainty.How to analyze models and economic problems under Knightian uncertainty becomes a challenging problem.Under the framework of probability spac e(?,F.P),the probability measure P and the linear expectation Ep are one-to-one correspondence.Therefore,the linear expectation cannot solve problems under probability uncertainties.Instead,we need to consider the nonlinear expectation which can describe a family of probabilities.In 1953,by nonadditivc measures,Choquet[G]proposed the famous capacity t heory and defined Choquet expectation.Choquet expectation is a nonlinear expectation and can be used to study model uncertainties in financial markets.For example,Schmeidler[87]introduced Choquet's expected utility theory.Chateauneuf et al.[3],Chen and Kulperger[5]et al.applied Choquet's integral to pricing for financial and insurance markers.But under the framework of Choquet expectation,it is impossible to define dynamically consistent conditional expectations.In 2004,Peng[72.73,74]first introduced the dynamically consistent nonlinear expectation space where ? is a given set,H denotes the space of random variables on ? and E denotes the nonlinear expectation on H.On(?,H,E),Peng defined the notions of G-normal distribution,maximum distribution,independence,and identical distribution and proved a series of results such as the law of large numbers and the central limit theorem under the nonlinear expectation.Then(?,H,E)becomes a theoretical framework comparable to the probability space(?.T.P).Compared with the linear expectation,one advantage of the nonlinear expectation theory is that it considers Knightian uncertainty and can solve more practical problems under model uncertainties.A typical nonlinear expectation is G-expectation EG which is induced by a nonlinear partial differential equation(G-heat equation).It can be represented by a family of probability measures {PO}??(?),i.e.,(?)Thus,G-expectation can be widely used to study probability uncertainties.For example,based on the nonlinear expectation,Epstein and Ji 20,21]obtained the asset pricing formula and utility model under volatility uncertainty.Eberlein et al.[22]extended the theory of two price(bid and ask price)economies to continuous time by the dynamically consistent G-expectation.Hu and Ji 37,38]obtained the stochastic maximum princi-ple and dynamic programming principle for stochastic optimal control problem under volatility ambiguity.Furthermore,in order to study stochastic processes under the nonlinear expecta-tion,Peng[75,76,78]introduced G-Brownian motion through G-heat equation and established the corresponding G-Ito 's stochastic calculus theory.Then many scholars studied G-Brownian motion and related topics.Gao[28],Lin[63],Lin[64]and Luo and Wang[67]et al.studied the stochastic differential equation driven by G-Brownian motion(G-SDE).Hu et al.[39,40]studied backward stochastic differential equations driven by G-Brownian motion(G-BSDE).By G-SDE and G-BSDE,we can generate a very rich class of stochastic processes,such as G-OU process and geometric G-Brownian motion.Besides,Hu and Peng 45]introduced G-Levy process and G-Poisson process by Levy-Khintchine representation under the nonlinear expectation.The stochastic pro-cesses defined above are temporal random fields with probability uncertainties and can be used to describe various models under volatility ambiguity in financial markets.In addition to economics,model uncertainties exist in all areas of the real world.For example,in biophysics,people usually assume that the random disturbance is a Gaus-sian white noise or empirical colored noise for molecular dynamic simulations.However,practical problems are not always described by these simplified models.Nonlinear white noise theory under model uncertainty may provide new tools to solve these problems.Therefore,we aim to extend Peng' s nonlinear expectation theory to physical fields and es-tablish a new nonlinear noise theory for studying various practical problems under model uncertainty.In order to introduce the nonlinear white noise or spatial-temporal white noise,we first need to establish a random field theory,including spatial random fields and spatial-temporal random fields,on the nonlinear expectation space.In particular,temporal randoin fields correspond to stochastic processes which have been introduced above.Under the framework of nonlinear expectation,time index and space index are two different parameters.We cannot use stochastic processes to describe spatial random fields as the independence under nonlinear expectation is directional,"Directional" means,if X and Y are two random variables,"X-is independent from Y" does not imply "Y is independent from X",Since spatial increments are undirected,spatial increments of the random field cannot be independent.Thus,both G-Brownian motion and G-Levy process are no longer suitable to describe space-indexed random fields,as they are incrementally independent.Based on this,Peng 80]introduced a nonlinear G-Gaussian process,whose finite-dimensional distributions are G-normally distributed.It is a spatial random field indexed by R.Unlike classical cases,G-Brownian motion is no longer a G-Gaussian process.For a random field defined on Rd or a more general parameter set,based on G-Gaussian process,we establish a G-Gaussian random fild and G-white noise theory under the nonlinear expectation.They can be used to describe spatial random fields under model uncertainty.To define a space-time-indexed random field,we treat the space parameter and time parameter separately,and then develop a new spatial-temporal G-white noise.In particular,the spatial-temporal G-white noise is a G-Brownian motion in time domain and a spatial G-white noise in space domain.Furthermore,we also extend G-Itos stochastic calculus theory to G-white noise and spatial-temporal G-white noise.Based on G-white noise theory,we consider stochastic partial differential equations under the nonlinear expectation.We prove the existence and uniqueness of the weak solution of the stochastic heat equation driven by spatial-temporal G-white noise and obtain estimates of the weak solution.In addition to spatial random fields,we also study the typical temporal random field,G-Brownian motion,under the nonlinear expectation.Unlike classical cases,the quadratic variation process of G-Brownian motion is no longer t,but a incrementally stable and independent process whose increments are maximum distributed.Based on this.Xu and Zhang[98,99],Lin[62]and Song[92]obtained Levy's martingale character-ization for 1-dimensional symmetry G-martingales.In this paper,we prove the martin-gale characterization theorem for general martingales,which may not be G-martingales,without the nondegenerate condition.Furthermore,we get the reflection principle for G-Brownian motion and G-Brownian motion.The reflection principle can be used to price barrier options under model uncertainty.Moreover,since the new spatial-temporal G-white noise is a G-Brownian motion in time omain,we can use Levy's martingale characterization theorem to study the quadratic variation process of the spatial-temporal G-white noise.The dissertation is organized as follows.In Chapter 1,we recall some basic notions and results of the nonlinear expectation theory.The notions of G-normal distribution,G-expectation and G-Brownian motion,and their basic properties are presented.In Chapter 2,we establish a general Gaussian random field and white noise the-ory under the framework of sublinear expectation.On the sublinear expectation space,by a generalized Kolmogorov's existence theorem,we first construct a G-Gaussian ran-dom field whose finite-dimensional distributions are G-normally distributed.G-Gaussian random field is random field defined on any parameter set and is suitable for describing random fields under model uncertainties.We then introduce a spatial type of G-white noise by defining a special family of generating functions under the sublinear expecta-tion.Stochastic integrals of functions in L2(Rd)with respect to the spatial white noise has also been established and this family of stochastic integrals is still a G-Gaussian random field.We emphasize that,within the framework of sublinear expectation,the space-indexed increments of G-white noise do not satisfy the property of independence.Furthermore,we focus on the spatial-temporal G-white noise on the sublinear expecta-tion space.Applying the similar method of G-Ito's integral,we develop the stochastic calculus with respect to the spatial-temporal white noise.Finally,taking the spatial G-white noise as an example,we induce the capacity theory and a pathwise description of G-white noise.In Chapter 3,we study stochastic heat equations driven by the spatial-temporal G-white noise.First,we extend Bochner integral and stochastic integral with respec-t to the spatial-temporal white noise,and prove the stochastic Fubini theorem on the extended space.This theorem guarantees that we can change the order of integration when verifying weak solutions.Then we give the definition and expression of the weak solution to the linear stochastic heat equation driven by spatial-temporal G-white noise.The uniqueness and estimates of the weak solution are also obtained.For the general quasilinear stochastic heat equations,we only consider equations satisfying the Lipschitz condition on the bounded space as the space of integrands is restricted.By Green's func-tion and Picard iteration,we obtain the existence and uniqueness of the weak solution and the corresponding estimates.In Chapter 4.we obtain Levy's martingale characterization and reflection principle of G-Brownian motion without the nondegenerat.e condition.In order to study symmetry martingales in the general sublinear expectation space,we first introduce the definition of consistent sublinear expeetation space and extend Peng's definition of stochastic calculus with respect to G-Brownian motion to a kind of martingales on the consistent sublinear expectation space.Then we consider general symmetric martingales,which may not be G-martingales,without the nondegenerate condition.For this general case,the traditional method can not be directly applied as the solution of G-heat equation may not be regular.We introduce a new kind of discrete product space method for martingales to deal with the degenerate case,and obtain the nondegenerate operator G? and G?-heat equation.The corresponding Levy's martingale characterization of G-Brownian motion is then proved by Taylor's expansion and approximation.In the symmetric G-martingale case,we apply a straightforward method by Ito's formula.Based on the martingale characterization,we obtain the reflection principle for G-Brownian motion by Skorohord's lemma and Ito-Tanaka's formula.However,we do not obtain Levy's martingale characterization for the more general G-Brownian motion.Instead,we give a discrete approximation of process(sgn(Bt))t?T and apply Krylov's estimate to get the coresponding reflection principle.In Chapter 5,we give the conclusions and outlooks for the future work.In the following.we present the main results of this dissertation.1.Gaussian random fields and spatial and temporal white noises under sublinear expectationIn this chapter,we establish a framework of G-Gaussian random field and G-white noise under the sublinear expectation.Furthermore,we extend G-Ito's stochastic cal-culus for G-Brownian motion to G-white noise.First,we present the definition and existence of G-Gaussian random field.Definition 1.Let(?:H,E)be a nonlinear expectation space and ? be a parameter set.An m-dimensional random,field on(?.H.E)is a family of random vectors W=(W?)???such that W???Hm for each ???.Let us denote the family of all sets of finite indices byDefinition 2.Let(W?)??? be an m-dimensional random,field on a sublinear expectation space(?,H,E).(W?)??? is called an m-dimensional G-Gaussian random field if for each?=(?1…,?n)?J?,(n × m)-dimensional random vector W(?),=(W?1,…W?n)is G-normally distributed.For each ?=(?1…,?n)?J?.we define GW?(Q)=1/2E[?QW?,W??],Q?S(n×m),where S(n × m)denotes the collection of all(n × m)x(n x m)symmetric matrices.Then(GW?)??J? is a family of monotone sublinear and continuous functions satisfying the consistency properties in the following sense:(1)Compatibility:For any(?1,…?n,?n+1)?J? and Q=(qij)i,j=1n×m?S(n × m),(?)(2)Symmetry:For any permutation ? of {1,…,n} and Q?(qij)i,j=1n×m?S(n×m).where the mapping ?-1:S(n×m)?S(n×m)is defined byTheorem 3.Let(G?)??J? be a family of sublinear and monotone functions such that.for each ?=(?1,…,?n)?J?,the real function G? is defined on S(n×m)?R.More-over,this family(G?)??J?r satisfies the same cornpatibility condition(10)and symmetry condition(11).Then there exists an m-dimensional G-Gaussian random field(W?)???on a sublinear expectation space(?,H,E)such that for each ?=(?1,…?n)?J?,W??(W?1,…,W?n)v is G-normally distributed,i.e.,GW?(Q)=1/2E[<QW?,W?>]=G?(Q),for any Q?S(n×m).Furthermore,if there exists another Gaussian random,field(W?)???,with the same in-dex set ?,defined on a sublinear expectation space(?.H.E)such that for each ??(?1,…,?n)?J?,W? is G-normally distributed with the same generating function.namely.W=W.then we have W=W.Based on G-Gaussian random fields,we introduce,the notion of G-white noise.Definition 4.Let(Q,H,E)be a sublinear expectation space and ??B0(Rd)?{A?B(Rd):?A<?},where ?A denotes the Lebesgue measure of A ? B(Rd).A 1-dimensional G-Gaussian random field W=(WA)A?? is called a 1-dimensional G-white noise if(1)For all A??,[WA2]=?2?A;(2)For each A1.A2 ? ?,A1 n A2=(?),we have where 0<?2<?2 are any given numbers.To prove the existence of G-white noise,by Theorem 3,it is sufficient to define an appropriate family of sublinear and monotone functions(G?)??J? such that G-Gaussian random field generated by(G?)??J? satisfies(1)and(2)in Definition 4.For each ?=(A1,…,An)?J?,we define a mapping G?(·):S(n)?R as follows:where G(a)=1/2?2a+-1/2?2a-for a?R.Then we present the existence of G-white noise through this special family of sublinear and monotone functions.Theorem 5.For each given sublinear and monotone function G(a)=1/2(?2a+-?2a-),a?R,let the family of generating functions(G?)??J? be defined as in(12).Then there exist-s a 1-dimensional G-Gaussian random field(W?)??? on a sublinear expectation space(?,H.E)such that,for each ?(A1,…,An)?J?,W??(WA1,…,WAn)is G-normally distributed,i.e.,(?)Furthermore,(W?)??? is a spatial G-white noise on(?,H,E),namely,conditions(1)and(2)of Definition 4 satisfied.If(W?)???r is another G-white noise with the same sublinear function G in(12),then W=W.For each p>1,we denote by LGP(W)the completion of H under the Banach norm?·?p:=(E[|·|p]1/p.Then(?.LGP(W),E)forms a complete sublinear expectation space.In the following,we always assume that G-white noise(W?)??? defines on(?,LG2(W),E).A very important property of spatial G-white noise is that its distribution is invari-ant under rotations and translations:Proposition 6.For each p?Rd and O?(?)(d):={O?Rd×d,OT?O-1},we set Tp.O{A)?{Ox+p:x?A}.for A ? B0(Rd).Then,for each A1,…An?B0(Rd),we have(WA1,…,WAn)=(WTp.O(A1),…,WTp.O(An)).Namelly the finite-dimensional distributions of W are invariant under rotations and translations.Denote L2(Rd)={f:?f?L22??Rd|f(x)|2dx??}.We can define stochastic integrals of functions in L2(Rd)with respect to G-white noise.First,for any simple function f(x)=(?)ai1Ai(x),n?N.a1,…,an?R,A1…An??,we define(?)By the following lemma,we can extend the stochastic integral to L2(Rd).The family of stochastic integrals is also a G-Gaissian random field.Lemma 7.If f:Rd?R is a simple function in L2(Rd),then E[|?Rdf(x)W(dx)|2]??2?f?L22.Theorem 8.Let ??B0(Rd)and W=(WA)A?? be a 1-dimensional G-white noise on a complete sublinear expectation space(?,LG2(W),E).Then {?Rd f(x)W(dx):f?L2{Rd)}is a G-Gaussian random field.In the following,we study a new type of spatial-temporal white noise which is a random field defined on the following index set?={[s,t)x A:0?s?t<?,A?B0(Rd)}.Specifically,we haveDefinition 9.A random field {W([s,t)×A)}([s,t)×A)?? on a sublinear expectation space(?,H,E)is called a 1-dimensional spatial-temporal G-white noise if it satisfies the following conditions:(1)For each fixed[s,t),the random field {W([s,t)×}A?B0(Rd)is a 1-dimensional spatial white noise that has the same family of finite-dimensional distributions as((?WA)A?B0(Rd);(2)For any r?s?t,A ? B0(Rd),W([r,s)×A)+W([s,t)x A)?W((r,t)×A);(3)W([s,t)×A)is independent of(W([s1,t1)×A1),…,W([sn,tn)×An)),if ti?s and A,Ai?B0(Rd),for i?1,…,n,where(WA)A?B0(Rd)is a 1-dimensional G-white noise.For each p?1,T?0.similar to G-Brownian motion and G-expectation.we can construct the canonical random field {W(s.t)x A):0?s<t?T,A?B0(Rd)},the space of random variables LGp(W[0,T)(resp.LGP(W))and the sublinear expectation E such that W is a spatial-temporal G-white noise under E.In the following,we define the stochastic integral with respect to the spatial-temporal G-white noise W.First,let Mp,O([O,T]× Rd)be the collection of simple random fields with the form:(?)where Xij?LGP(W[O,ti]),i?0,…,n-1,j=1,…,m,0?t0<t1<…<tn=T,and {Aj}j=1m ?B0(Rd)is a mutually disjoint sequence.The Bochner integral of f?Mp,O([0,T]× Rd)with the form as(13)is defined by(?)It is clear that IB:Mp,O([O,T]×Rd)?LGp(W[0,T])is a linear mapping.For a given p>1,denote by MGP([O,T]× Rd)the completion of MP,O([0,T]x Rd)under the norm?·?Mp:=(E[?0T fRd|·|pdsdx])1/p.The stochastic integral with respect to the spatial-temporal white noise W from M2,0([0,T]x rd)to LG2(W[0,T)can be defined as follows.(?)Then we have the following lemma which allows us to extend the domain of the stochastic integral to MG2([0,T]x Rd).Lemma 10.For each f?M2,0([O,T]× Rd),we have E[?0T?Rdf(s,x)W(ds,dx)]=0,E[|?0T?Rdf(s,x)W(dx,dx)|2]??2E[?0T?Rd|f(s,x)|,dsdx].Therefore,we can continuously extend the domain M2,0([0,T)× Rd)of the stochastic integral(14)to MG2([0,T]x Rd).Indeed,for any f?MG2([0,T]x Rd),there exists a sequence of simple random fields fn ? M0([0,T]× Rd)such that lim ?fn-f?M2?0.Then we can define(?)2.Stochastic heat equations driven by the spatial-temporal G-white noiseIn this chapter,we study the weak solution of the stochastic heat equation driven by spatial-temporal G-white noise.First,we have the stochastic Fubini theorem under sublinear expectation.For each T>0,let M2,0([0,T]2 x R2)be the collection of simple random fields in the following form:(?)where {t0,…,tn} and {s0,…,sn},are any given partitions of[0,T],{Aj}j=1m and{BK}k=1m(?)B(Rd)are two mutually disjoint sequences,and Xijlk?LG2(W[0,ti]),for i,l=0,…,n-1 and j.k?1,…,n.Denote by MG2([0.T]2×R2)the completion of M2.0([0,T]2 × R2)under the norm ?·?M2=(?0T?R?0TE[|f(t,x,s,y)|2]dxdtdyds1/2.Then we obtain the following equality of changing the order of integration.Theorem 11(Stochastic Fubini Theorem).For each T>0,f(t.x.s.y)?M2.0([0.T]2 ×R2),we have?0T?R[?0T?Rf(t,x,s,y)W(dt,dx)]dyds=?0T?R[?0T?Rf(t,x,s,y)dyds]W(dt,dx).Furthermore,if f(t.x,s,y)?MG2([0,T]2 x R2),then for each K?B0(R),we have?0T?K[?0T?Rf(t,x,s,y)W(dt,dx)]dyds=?0T?R[?0T?Kf(t,x,s,y)dyds]W(dt,dx).In particular,it is easy to verify the following corollary.Corollary 12.For each f ? MG2([0.T]2×R2),K?B0(R),we have?0T?K[?0s?Rf(t,x,s,y)W(dt,dx)]dyds=?0T?R[?tT?Kf(t,x,s,y)dyds]W(dt,dx).For each T>0,set ?=[0.T]×R.We consider the spatial-temporal C-white noise {W(t,x)W([0.t)×[x?0.0?x):t?[0.T],x×?R}.still denoted by W?{W(t,x):t?[0.T].x?R}.Then W(t,x)is nowhere-differentiable in the ordinary sense,but its derivatives will exist in the sense of Schwartz distribution.Define W(t,x):=(?)2W(t,x)/(?)t(?)x,t?[0,T],x?R,which means that for each test function ??Cc?(R2),?0T?RW(t,x)?(t,x)dxdt=?0T?RW(t,x)(?)?(t,x)/(?)t(?)x(dt,dx).where Cc?(R2)denotes the space of infinitely differentiable functions with compact sup-ports in R2.It is easy to prove the following proposition.Proposition 13.Let {W(t,x):t ?[0,T]·×R} be the derivative of {W(t,x):t?[0,T],x ?=R} in the sense of(15).For each function ? ? Cc?(R2).we have?0T?RW(t,x)?(t,x)dxdt=?0T?R?(t,x)W(dt,dx).By the above proposition,we can give the definition of the weak solution to the stochastic heat equation.Consider the following linear stochastic partial differential equation driven by spatial-temporal G-white noise:(?)where U0:R?R is the initial function which is nonrandom and Borel measurable.Definition 14.Let u0(x)?S2(R),i.e.,supx?R|u0(x)|2<?.{u(t,x):(t,x)?[0,?)×R]is said to be a weak solution of stochastic heat equation(16)if(1)For each T? 0 and K ? B0(R),(u(t,x)0<t<T.x?K ?SG2([0.T]× K):(2)For each T? 0 and ?(t,x)?Cc?(R2),(u(t,x)0?t?T,x?R satisfies the equation?Ru(T,x)?(T,x)dx-?Ru0(x)?(0,x)dx=?0T?Ru(t,x)((?)?(t,x)/(?)t+(?)2?(t,x)/(?)x2)dxdt+?0T?R?(t,x)W(dt,dx),where SG2([0,T]× K)denotes the completion of M2,0([0.T]x K)under the norm ?·?s2?supt?[0,T]supx?K(E[|·|2]1/2.Then we can prove the existence and uniqueness of the weak solution of the linear stochastic heat equation(16).Theorem 15.If u0(x)?S2(R),then the linear stochastic heat equation(16)has a unique weak solution.The explicit expression of the weak solution is as follows:u(t,x)=?Ru0(y)p(t,x-y)dy+?0t?Rp(t-x,x-y)W(ds,dy),(?)t>0,x?R,where p(t,x)=1/(?)e-x2/4t,(?)t>0,x?R.Proposition 16.Let u0(x)? S2(R)and {u(t,x):(t,x)E[0.?)x R} be the weak solution of linear stochastic heat equation(16).Then for any 0<a<1,(t,x)?(0,?)x R,there exists a constant C such that for h?0?the following properties hold:(1)E[|u(t,x+h)-u(t,x)|2]?Ch?;(2)E[lu(t+h,x)-u(t,x)|2]?C(?).For each given constants T>0.K>0,we consider the following quasilinear stochastic heat equation driven by spatial-temporal G-white noise:(?)where a(x)?Cb,Lip(R),u0(x)?S2([0,K}).Definition 17.For any given T?0,K?0,let u0(x)? S2([0,K]),a(x)?Cb.Lip(R).For each test function?(t,x)? Cc?(R2)such that)=(?)/(?)x?(t,K)=0,t?T,if(u(t,x))0?t?T,0?x?K?SG2([0,T]×[0,K])satisfies the equation(?)(18)then {u(t,x):(t,x)?[0.T]×[0.K]} is called a weak solution of(17).By Green 's function and Picard iteration,we obtain the existence and uniqueness of the weak solution of the quasilinear stochastic heat equation driven by spatial-temporal G-white noise.Theorem 18.If u0(x)? S2([0.K])and a(x)? then the quailinear stochas-tic heat equation(17)has a unique weak solution {u(t,x):(t,x)?[0,T]×[0,K]} ?SG2([0,T]×[0,K]).Furthermore,we have the following estimates.Proposition 19.Let u0(x)? S2([0,K]),a(x)? Cb.Lip(R)and {u(t,x):(t,x)?[0,T]×[0,K]}?SG2([0,T]×[0,K?)be the weak solution of(17).For any 0<a<1,there exists a constant C such that for each x.y ?[0,K],t.s ?[0,T],the following properties hold:(1)E[|u(t,x)-u(t,y)[2]?C|x-y|?;(2)E[|u(t.x)-u(s,x)|2]?C|t-s|1/2.3.Levy s martingale characterization and reflection principle of G-Brownian motionIn this chapter,we consider Levy 's martingale characterization for general symmet-ric martingales on the consistent sublinear expectation space.Based on this characteri-zation,we prove the reflection principle of G-Brownian motion and G-Brownian motion.Under the framework of sublinear expectation,we first introduce the definition of the consistent sublinear expectation space.Let Q be a given set and(Ht)t?0,R/be a family of linear spaces of real-alued functions on ? such that(H1)H0=R and-Hs(?)Ht(?)H for each 0 ?S?t;(H2)If X ?Ht(resp.H),then |X| ?Ht(resp.H):(H3)If Xi,...,X,? Ht(resp.H),then(?)(Xi,...,Xn)? Ht(resp.H)for each(?)?Cb.Lip(Rn).Definition 20.A consistent sublinear expectation on(Ht)t?0 is a family of mappings Et:H?Ht,t?0,which satisfies the following properties:for each X,Y?H,(1)Monotonicity:X>Y implies Et[X]?Et[Y];(2)Constant preserving:Et[?]=?for??Ht:(3)Sub-additivity:E[X+Y]?Et[X]+Et[Y];(4)Positive homogeneity:Et[?X]??Et[X]for each positive and bounded ? ?Ht;(5)Consistency:Es[Et[X]]=Es[X]for s?t.The triple(?,H,(Ht)t?0,(Et)t?0)is called a consistent sublinear expectation space and Et is called a sublinear conditional expectation.For each given p>1,we denote by(resp.LP(?))the completion of Ht(resp.H)under the norm ?·?p:=(E[|·|P])1/P.Then(?,L1(?),(L1(?t))t?0?(Et)t?0)forms a consistent sublinear expectation space,which is called a complete consistent sublinear expectation space.The following properties are very useful in sublinear analysis.Proposition 21.Let(?,L1(?).(L1(?t))t?0,(Et)t?0)be a complete consistent sublinear expectation space.Assume X?(L1(?t))n and Y ?(L1(?))m.Then Et[(?)(X,Y)]=Et[(?)(x,Y)]x=x for each(?)?Cb.Lip(Rn+m).In particular.for each bounded ??L1(?t),Et[?Z]=?+Et[Z]+?Et[-Z]for each Z?L1(?).Now we give the definition of martingales on the consistent sublinear expectation space.Definition 22.Let(L1(?t))t?0,(Et)t?0)be a complete consistent sublinear expectation space..A d-dimensional adapted process is a family of random vectors(Xt)t?0 such that Xt E(L1(Qt))d for each t?0.Definition 23.Let(L1(?t))t?0,(Et)t?0)be a complete consistent sublinear expectation space.A d-dmensional adapted process Mt=(Mt1,...,Mtd)T,t?0,is called a martingale if Es[Mti]=Msi for each s?t and i?d.Furthermore,a d-dimensional martingale M is called symmetric if E[Mti]=-E[-Mti]for each t?0 and i?d.By introducing a new kind of discrete product space method for martingales,we obtain the following Levy's martingale characterization of G-Brownian motion on the complete consistent sublinear expectat.ion space.Theorem 24.Let(?.L1(?),(L1(?t))t?0,(Et)t?0)a complete consistent sublinear ex-pectation space and G:S(d)?R be a given monotonic and sublinear function.Assume(Mt)t?0 is a d-dimensional symmetric martingale satisfying M0=0,Mt ?(L3(?t))d for each t?0 and sup{E[|Mt+?-Mt|3]:t?T}=o(?)as ?? 0 for cach fixed T>0,Then the following conditions are equivalent:(1)(Mt)t?0 is a G-Brownian motion;(2)Et[|Mt+s-Mt|2]?Cs for each t,s?0 and the process 1/2tr[A<M>t]-G(A)t,t?0,is a martingale for each A ? S(d),where C>0 is a constant;(3)The process 1/2<AMt,Mt>-G(A)t,t?0,is a martingale for each A ? S(d).We now present Levy's martingale characterization of G-Brownian motion on G-expectation space.In this case,we do not need the assumptions Mt ?(L3(?t))d and sup{E[|Mt+?-Mt|3]:t?T}=o(?)as ? ? 0 stated in the above theorem.Theorem 25.Let G:S(d')?and G:S(d)?R be two given monotonic and sublin-ear functions,and let(?,LG1(?),(LG1(?t))t?0,(Et)t?0)be a G-expectation space.Assume(Mt)t?0 is a d-dimensional symmetric martingale satisfying M0?0,Mt?(LG2(Qt))d for each t>0.Then the following conditions are equivalent:(1)(Mt)t?0 is a G-Brownian motion;(2)The process 1/2<AMt,Mt>-G(A)t,t?0,is a martingale for each A ? S(d);(3)Et[|Mt+s-Mt|2]?Cs for each t,s?0 and the process 1/2tr[A<M>t]—G(A)t,t?0,is a martingale for each A ? S(d)7 where C>0 is a constant.Based on the martingale characterization,now we consider the reflection principle of G-Brownian motion on G-expectation space.Let ??C0([0,?))and(Bt)t?0 be the canonical process on ?.For each t? 0,set Lip(?t)={(?)(Bt1,Bt2-Bt1,…,Btn-Btn-1):n ? N.0?t1<…<tn?t,(?)?Cb.Lip(Rn)},and(?)For each given 0?a2??2 with ?2>0,define G(a):1/2(?2a+-?2a-)for a ? R,Peng in[78]constructed a sublinear G-expectation EG[·]on Lip(?),under which(Bt)t?0 is a 1-dimensional G-Brownian motion.First,we have the following Ito-Tanaka formula by the representation theorem for G-expectation.|Bt-a|=|a|+?0t sgn(Bs-a)dBs+Lt(a),q.s.,where Lt(a)is called the local time in a of G-Brownian motion B under EG.By the following lemma and the well-known Skorokhod lemma,we can prove the reflection principle for G-Brownian motion.Lemma 26.Let(Bt)t?0 be a 1-dimensional G-Brownian motion.Then ?0t sgn(Bs)dBs,t?0,is still a G-Brownian motion.Theorem 27.Let(Bt)t?0 be a 1-dimensional G-Brownian motion and(Lt(0))t?0 be the local time of B under EG[·].Then(St-Bt,St)t?0=(|Bt|,Lt(0))t?0 under EG[·],where St=sups?t Bs for t?0.Similarly,let G:R?R be a nonlinear functional dominated by G,Peng con-structed a nonlinear G-expectation EG[·]on Lip(?),under which the canonical process(Bt)t?0 is a process with stationary and independent increments.(Bt)t?0 is called a 1-dimensional G-Brownian motion under EG.Then we have the following reflection principle of G-Brownian motion.Theorem 28.Let(Bt)t?0 be a 1-dimensional G-Brownian motion and(Lt(0))t?0 be the local time of B under EG[·].Then(St-Bt,St)t?0=(|Bt|.Lt(0))t?0,under EC[·],where St=sups?t Bs for t?0.