Research About G-expectations And Related Questions

Superhedging, uncertainty problem and measures of risk casued a great attention in finance since the pioneer work of Artzner, Delbaen, Eber and Heath [1]. This is also the starting point of a new theory of stochastic calculus which gives us a new insight to characterize and calculate varies kinds of financial risk. Motivated by this, Peng has introduced a new notion of nonlinear expectation, the so-called G-expectation, which derives from a nonlinear heat equation. Together with the notion of G-expectation, Peng [36,37,38] also introduced the related G-normal distribution, G-Brownian motion and the stochastic calculus with respect to G-Brownian motion. The G-expectation is a sublinear expectation, which is related to risk measuresρin finance by the relationρ(X):= E[-X], where X runs the class of contingent claims. Although G-expectations represent only a special case, their importance inside the class of nonlinear expectations reflects in the law of large numbers and central limit theorem under nonlinear expectation [40,41].This dissertation focuses on the research about G-expectation and the related questions. Firstly, we recall that g-expectation is the solution of a backward stochastic differential equation (BSDE), which is completely de-termined by the generator g. Compared with g-expectation, G-expectation is also totally determined by the nonlinear heat equation G, so we regard the function G to be the generator of G-expectation. In this dissertation, we first consider the comparison and the converse comparison theorems of G-expectations, and then we present the relationship between the operations of G-expectations with the operations of the generators G. The notion of G- distribution as an extension of G-normal distribution was first introduced by Peng [40]. It generalizes the G-normal distribution in the sense that mean-uncertainty can also be described. Based on the notion of G-distribution [40], we introduce a new notion of nonlinear expectation named G-distributed ex-pectation, we then extend our results in the framework of G-expectation to a more general case. The last chapter studies the existence and uniqueness of the solution to stochastic equations driven by G-Brownian motion with integral coefficients.This dissertation consists of four chapters, whose main contents are de-scribed as follows:In chapter one, we first get the comparison and the converse comparsion theorem of G-expectations:Theorem 1.3.1 Let Ei[·], i=1,2 be two G-expectations on the space (Ω,H0) with each driver Gi=(?) defined by (1.1), respectively, then the following are equivalent: 1,(?) 2,(?)Remark:The corresponding theorem for multi-dimensional case is an easy consequence of Proporsition 3.3.1 and 3.3.2.We then present the relationship between the operations of G-expecations with the operations of the drivers G, more precisely, (?) with (?)and (?) with (?) . Chapter one gives the answer that:Proposition 1.3.2 (?) is the smallest sublinear expectation on the space (Ω,H0) larger than E1 and E2.If either (?)or (?), then E1 V E2[·] and (?) are two G-expectations with each driver (?) and (?) respectively, otherwise, (?) is not a G-expectation, while (?) is not a sublinear expectaion. In chapter two,we focuse on the following optimization problem:This model is called the optimal risk transfer model when G-expectations are replaced by risk measures.The convex risk measure case was first stud-ied by El Karoui and Barrieu[2,3,4],in particular those described by g-expectation.We now define in the same way that The main aim of this chapter is to present the relationship between the above introduced operator E1口E2[.]and the G-expectation EG1□G2[.].More pre-cisely, we have the following theorem:Theorem 2.3.2 Let E1[.]and E2[.]be the two G-expectations on the space (Ω,H0),which have been defined above.If G1□G2(.)≠-∞,then E1口E2[.] is a G-expectation on(Ω,H0)and has the driver G1□G2,i.e.,Corollary 2.3.10 Let 0≤σi≤σi,1≤i≤n,and denote by Ei[.] the (?) expectation on the space(Ω,H0).Then under the assumption also is a G-expectation and has the driver (?).Moreover,for any permutation i1,...,in of the nat-ural numbers 1,...,n it holds:Remark:If (?) is empty, then otherwise (?) is a (?) expectation,where (?)We then study the continuity and dynamical properties of this inf-convolution problem. Firstly, we extend the definition of E1口E2[·]from Lip0 to LG11,that is Now we have the following two statements:Proposition 2.3.11 We suppose that(?)such that for some (?)then that isProposition 2.3.12 suppose that for a fix random variable x∈LG11,there is a random sequence (?),such that then for every t>0,(?) is the random sequence such thatCombining with the representation theorem of G-exepctation in Denis et al.[10] and Hu,Peng[25],we give the probabilistic interpretation of this inf-convolution problem:Theorem 2.4.2 For all(?),then where Pθis the law of (?),t≥0.In chapter three,based on the notion of G-distribution,following the way of Peng[38],we first introduce the notion of G-distributed expectation, and the related G-distributed process({B}t,bt}t≥0).We then study the inf-convolution problem in G-distributed expectation sapce,and we no longer constrain ourselves to one dimensional case any more. At the beginning,we give the characterization theorem of G-distributed process:Theorem 3.3.3 Let(Bt,bt)t≥0 be a 2d-dimensional random vector process defined on a sub-expectation space(Ω,H,E)such that (i)B0=0,b0=0；(ii)For each t,s≥0,the increment(Bt+s,bt+s)-(Bt,bt) has the same dis-tribution as(Bs,bs)and is independent of{(Bt1,bt1),(Bt2,bt2),…,(Btn,btn)}, for all n≥l and 0≤t1,…,tn≤t. (iii)E[Bti]=E[-Bti]=0,and limt↓0 (?),limt↓0 (?), where Bti is the i-th dimension of Bt,i=1,…,d.Then(Bt,bt)is a G-distributed process withFollowing the idea of chapter two,we show that under the sublinear ex-pectation E1口E2[.]the canonical process(Bt,bt)t≥0 satisfies the assumptions of Theorem 3.3.3 for G1□G2.This has as consequence that(Bt,bt)t≥o is a G1□G2-distributed process under疤1口E2[.],and implies thatE1口E2[?]= EGl□G2[.].More precisely,we have:Theorem 3.3.4 Denote by E1[.] and E2[.]two G-distributed expectations On the space(Ω,H0)with each driver G1 and G2 respectively,if G1□G2(.)≠-∞,thenE1口E2[?]also is a G-distributed expectation on(Ω,H0)and has the diver G1□G2,i.e.,E1口E2[?]=EG1□G2[?].This theorem generalizes the results of chapter two in the sense that mean—uncertainty can also be decribed. In the sequel,we give some Corol-laries,for example:we define from Hu and Peng[24],we know that{Bt}t≥o is a generalized G-Brownian motion,and the function is the unique viscosity solution of the following PDE: then from Theorem 3.3.4, we have the following Corollary immediately:Corollary 3.4.4 If (?), (Bt)t≥0 is also a generalized G-Brownian motion under the G-distributed expectation E1□E2...□En. and has the driver G1□G2...□Gn.Chapter four studies the existence and uniqueness of the solution to G-SDEs and G-BSDEs with non-Lipschitz conditions, whose main results are the following three theorems:Theorem 4.3.1 Under the assumptions: where (?) is the Hilbert-Schmidt norm of a matrix A=(aij),β1(·)∈MG2(0,T), (?),(?) are two square integrable real value functions, and p:(0,+∞)→(0,+∞) is continuous, increasing, concave function satisfying then there exists a unique continuous process X(·,x)∈MG2([0,T];Rn) such thatFurthermore, we consider the existence and uniqueness of a solution to the stochastic differential equation under some weaker conditions.Theorem 4.3.2 We suppose the following condition:for any x1, x2∈Rn where (?) is p-integrable, p> 2,β:[0, T]→R+is p-Lebesgue integrable, and pi, p2:(0,+∞)→(0,+∞) are continu-ous, concave and increasing, and both of them satisfy ((?))..Furthermore, we assume that is also continuous, concave and increasing, and Then there exists a unique solution X. in MGp([0, T];Rn) such thatRemark:For the functions which satisfy the assumptions of Theorem 4.3.1 and 4.3.2, the reader is referred to [26,45] and Chapter 4 of this dissertation.The last section gives the existence and uniqueness of a solution to G-backward stochastic differential equation with integral-Lipschitz coefficients.Consider the following type of G-backward stochastic differential equa-tion (G-BSDE): whereξ∈LG1(FT;Rn), and f,gij are given functions satisfying f(·,x), gij(·, x)∈MG1([0, T];Rn) for all x∈Rn and i, j= 1,..., d. We assume further that, for all y, y1and y2∈Mn, where c> 0,β∈MG1([0, T]; R+) and p:(0,+∞)→(0,+∞) is a continuous, concave, increasing function satisfying ((?)).Theorem 4.4.1 Under the assumptions above, (*) admits a unique solution Y∈MG1([0,T],Rn).