| G-Brownian motion and G-expectation in finite dimensional spaces was firstly introduced by using a nonlinear partial differential equation called G-heat equation in Peng [36], starting with which and up to the present date, various aspects of G-expectation have been studied:Denis-Hu-Peng [15] and Hu M.-Peng [24] proved that G-expcctation admits a representation with respect to a weakly compact set of proba-bility measures; Gao [19] and Gao-Jiang [20] studied, respectively, pathwise properties and large deviations for stochastic differential equations driven by G-Brownian motion; Soncr-Touzi-Zhang [45], Hu Y.-Pcng [25] and Peng-Song-Zhang [40] studied the rep-resentation theorem for G-martingales and so on. The notion of G-cxpcctation has received very strong attention because of its several motivations and applications, es-pecially for, among others, economic/financial problems with volatility uncertainty and numerical methods for high dimensional fully nonlinear PDEs.In classical theory, it is well known that Levy processes, which generalize Brownian motion by allowing discontinuous paths while preserving independent and stationary increments, arc at the same time simple enough to study and rich enough for applica-tions, or at least to be used as building blocks of more realistic models. Hu M.-Peng [23] introduced finite dimensional G-Levy processes and studied their corresponding distributions which satisfy a new type of nonlinear integro-PDEs.This dissertation focuses on the research about G-Levy processes and the related problems. This dissertation consists of three chapters, whose main contents arc de-scribed as follows:(Ⅰ) In Chapter1, we are concerned with the representation of the sub-linear expectation EG[·] associated with the new stochastic process G-Levy processes. We show the existence of weakly compact family of probabil- ity measures Ρ with respect to which EG[·] can be represented and give in further some characterizations of the space LG1(Ω). A generalized version of Kolmogorov-Chentsov’s criterion for cadlag modification of a stochastic process is also obtained.Denis-Hu-Peng [15] and Hu M.-Peng [24] showed that the G-expectation associ-ated with a G-Brownian motion can be represented as the "pointwise" maximum over a relatively compact family of probability measures. Since G-Levy processes contain G-Brownian motion as a special case, one natural problem is whether the sublinear ex-pectation EG[·] associated to G-Levy processes has a similar representation? Motivated by the two methods in Denis-Hu-Peng [15] and Hu-Peng [24], we give an affirmative answer to this question in this chapter.In chapter1, the state space of our study is Ω=D([0,∞), Rd) instead of the space C([0,∞),Rd) in [24], where D>([0,∞),Rd) denotes the space of all Rd-valued cadlag functions on [0,∞). Then Ω equipped with a Skorohod distance d°is a Polish space. Let (Bt)t≥0denote the G-Levy process associated to a sublinear expectation E. For any T>0, let Lip(ΩT) and Lip(Ω) denote, respeetively, the spaees of finite dimensional cylinder random variables: Lip(ΩT):=｛φ(Bt1ΛT, BtΛ2T,…, BtnΛT):n∈N, t1,t2,…,tn∈[0,∞), φ∈Cb,Lip(Rd×n)｝ Lip(Ω):｛φ(Bt1,Bt2,…,Btn):n∈N,t1,t2,…,tn∈[0.∞),φ∈Cb,Lip(Rd×n)｝ Then for p≥1. let LGp(ΩT)(resp. LGp(ΩT)) be the topological completion of Lip(ΩT)(resp. Lip(Ω) under the Banach norm‖·‖p:=E[|·|p]1/p.LetΩ=(Rd)(0,∞) denote the space of all Rd-valued functions (ωt)t≥0and:(?)(Ω) denote the σ-algebra generated by all finite dimensional cylinder sets. Let (Bt)t≥0denote the corresponding canonical process. The spaces Lip(Ω) and Lip(ΩT) are defined in a similar way as Lip(ΩT) and Lip(Ω). Then we can construct a sublinear expectations E on (Ω,Lip(Ω)) such that (Bt(ω))t≥0is a G-Levy process.Firstly, based on the elementary representation of sublinear expectation introduced in [39], we can find a family of probability measures (?), on (Ω,(?)(Ω)) such thatThen we give a generalized Kolmogorov-Chentsov’s criterion for the existence of a cadlag modification of a process with respeet to capacities a.s followsTheorem1.3.10(Kolmogorov-Chentsov’s Criterion) Let (Xt)t∈[0,1] be a real val-ued stochastie process such that for all t∈[0,1], Xt belongs to L1(Ω) with respeet to an upper expectation E. If it satisfies the following conditions:(i)(χt)t∈[0,1] is separable and for any t∈[0,1], there exists a>0, such that lims'E[|χs-χt|α]=0;(ii) for some C,r>0, p, q≥0with p+q>0, and all0≤s≤u≤t≤1, it holds that E[|χt-χu|p|χu-χs|q]≤C|t-s|1+r then it admits a cadlag modification.A tightness criterion in Jacod and Shiryacv [29] is adapted to get a main result of this chapter:Theorem1.6.4(Kolmogorov-Chentsov’s criterion for weak relative compact-ness) Let (?) be any subset of the collection of all probability measures on Do([0, T],R) and E the upper expectation related to&. If the following conditions are satisfied:(i)(?)a>0, such that E[|χt-χs|]≤C|t-s|α,(?)t,s∈[0,T];(ii) E[|χt-χu|p|χu-χs|q)≤C|t-s|1+r, for some C.r>0,p,q≥0with p+q>0, and all0<s <u <t <T, then&*is relatively compact.Theorem1.3.13For each monotonic and sublincar function Gχ [f(·)]: Rd'R, where f∈Cb3(Rd) with f(0)=0, let EG be the corresponding sublincar expectation on (Ω,Lip(Ω)). Then there exists a relatively compact family of probability measures (?)1:=｛QoB-1:Q∈(?)e｝ on (Ω, Lip(EG)) such that where B is the cadlag modification of B.As the second main part of this chapter, we then construct a concrete weakly rela-tively compact family of probability measures through t he method of optimal stochastic control.Proposition1.4.4where Pe is the law of the process Bt0,θ, t≥0, for θ∈A0.∞θ and A0.∞θ:=｛θ=(θc,θd)=(1θc,2θc,θd): θc is9c-valued F-adaptcd and θd is θd-valued F-predictablc｝ with θ=(θc, θd)=(1θc,2θc, θd), a given bounded and closed subset in Rd×3d.Significantly, the fundamental difference between the function spaces related to G-Brownian motion and that to G-Levy process is as result of the difference of the state spaces, which has a consequence that Lip(Ω)(?)Cb(Ω) in our case. We can nevertheless give the following statement:Proposition1.5.2For each Χ∈Lip(Ω) and ε>0, there exists Y∈Cb(Ω) such that EG[|X-Y|]<ε.Then we can further imply that LlG{il) CL). In the sequel, we can give the represen-tation formula for Efl<associated to the (7-Levy process, more precisely, we have:Theorem1.5.4For each Χ∈LG1(Ω), we have EG[Χ]=E(?)[Χ]=E(?)1[Χ], where&is the closure of2PX under the topology of weak convergence.(Ⅱ) In Chapter2, we study the large deviations for G-Levy process with sample paths of bounded variation and obtain a representation of the rate function.In classical theory, the two basic building blocks of every Levy process are the Brownian motion (the diffusion part) and the Poisson process (the jump part). We know that Brownian motion has continuous paths whereas a Poisson process does not. On the other side, a Poisson process is a non-decreasing process and thus has paths of bounded variation over finite time horizons, whereas a Brownian motion has paths of unbounded variation over finite time horizons. Under the framework of G-expectation, a joint large deviation principle for G’-Brownin motion and its quadratic variation process which gives in further the large deviation principle for stochastic differential equations driven by G-Brownian motion is obtained by Gao and Jiang [20]. The aim of this chapter is to present a large deviation principle for a special class of G-Levy processes, namely the processes with sample paths of bounded variation.Let Dun Dc and Ds denote, respectively, the space D0([0,1]. Rd) with the weak topology, uniform topology and Skorohod topology d°. For (?)>0, G(f) and S,(f) are, respectively, the e-neighbourhoods of f in Dt. and Ds. Let X be a topological space. Denote by the logarithmic moment generating function associated with E (For simplicity we use E instead of E6’ in the whole chapter). Let (δ-,δ+) be the interval such that EeδX(1)<∞for δ∈(δ-,δ+) and EeδX(1)=∞for δ∈[-∞,δ+) U (δ+,∞]. Following the ideas of [35], combining with the results of Chapter1. especially Theorem1.5.1. we can obtain several main results in this chapter as follows: Theorem2.4.7If|δ±|>0, then I(x) is a good rate function in Ds. Let (Xε, ε>0) be a family of measurable maps from Ω into a Polish space (X, d).Theorem2.4.16Let F be a weakly closed subset of M[0,1], thenTheorem2.4.17Let G be a weakly open subset of M[0,1]. ThenTheorem2.3.5Let (X(t))t≥o be a G-Levy process with sample path of bounded variation and assume thatThen (C(εX(t/ε)|t∈|O,T]∈·),ε>0) satisfies the LDP on Ds with speed λ(ε)=ε and rate function/(·), where where AC denotes the space of absolutely continuous functions.Finally, we can apply our result to G-Poisson process as an example of G-Levy processes, and obtain its corresponding rate function as follows: where Hence the rate function is different from that of classical Poisson process due to the uncertainty of intensity0≤μ1≤μ≤μ2. (Ⅲ) In Chapter3, we first introduce G-Levy processes in infinite dimen-sional space, establish the Levy-Ito integrals and introduce the Ornstcin-Uhlenbeck processes of which we study the connection to a new type of fully nonlinear integro-PDEs.Despite of increasing popularity of theory of G-expcctation, the most part of results are dedicated to the finite dimensional ea.se, so how to generalize the classical infinite di-mensional stochastic processes in a general framework of sublinear expectation becomes an interesting problem. Recently, the theory of G-expcctation and related stochastic calculus with respect to G-Brownian motion in infinite dimensions have been firstly obtained in [26]. Such process is of fundamental importance as the generalizations of finite dimensional real valued G-Brownian motion.Let H denote the real separable Hilbert spaces equipped with the inner product (∵)H, and the norms||·||H. Let A be linear, densely defined maximal monotone operator on H. Suppose there exists a bounded, linear, positive, self-adjoint operator on H denoted by B such that A*B is bounded on H and ((A*B+C0B)x.x)≥0, for all x∈H for some C0≥0. This condition is referred to as weak B-condition. Then we define the space H-1to be the completion of H under the norm||x||-1=||B1/2||H.H-1is a Hilbert space interpreted a.s the dual of H=B1/2H which is equipped with the inner product (x,y).1=(B1/2x.B1/2y). We first introduce a new type of integro-PDEs as follows: where A:D(A)'H is a generator of G(rsemigroup and G:H×S(H)×UCb1.2((0,T)×H)'B is defined by and give the following statement:Theorem3.4.13(Comparison theorem) Let the functions u∈BUC([0,T]×H-1) and u∈BUC([0,T)×H-1) be, respectively, a viscosity subsolution and a viscosity supersolution of above equation. Suppose that assumptions (A1)-(A5) are satisfied. If u(T,x)≤v(T,x) for all x∈H, then u≤v (i.e., u(t·)≤v(t,·) for all t>0).Theorem3.5.4(Ito’s isometry inequality) For stochastic integral/:HMG1,(d)(0, T)'HK1.(0, T), the next inequality holds, Let (Bt)t≥0be a G-Lcvy process taking values in a separable Hilbcrt space H. Set XTt,x=e-(T-t)Ax+∫tTe-(T-σ)AdBσ, which is defined as an Ornstein-Uhlenbeck process with respect to Bt, then we have the following main result of this chapter:Theorem3.6.5Let A be an infinite dimensional generator of Co-semigroup satisfy-ing the condition of weak B-case. Let Bt be a G-Levy process with the correspond-ing sublinear expectation E, Xt is a corresponding Ornstein-Uhlenbeck process, then u(?)=u(t,x)=E[(?)(XTt.x)] is the unique viscosity solution of above described intergro-PDEs:... |
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