Properties Of Nonlinear Expectations And Their Applications In Financial Risks

By Pardoux and Peng [54], we know that there exists a unique adapted and square integrable solution to a backward stochastic differential equation (BSDE for short) of the type provided the function g is Lipschitz in both variables y and z, andξand (g(t,0,0))o1, if E[|ξ-ηn|p|Ft]→0, a.s., t∈[0,T], thenTheorem 1.3.1. Assume (ⅰ)-(ⅲ) hold. (a) Then the limit (yt) of (ytn) has a form: where (gt0) is the weak limit of (gtn) in Lp(0,T;P; R), (zt) is the weak limit 0/(ztn) in Lp(0,T; P; R), (At) is an RCLL increasing process with A0=0 and E[(AT)p] <∞. (b) Furthermore, in addition of (ⅰ)-(ⅲ), we assume (ytn) uniformly converges to (yt) with respect to t, then, for any p'∈(0,p), (ztn) strongly converges to (zt) in Lp' (0,T;P;R), Theorem 1.3.2. Assume that the terminal condition {ynT),the function g and the increasing process (Atn) satisfy (A.1'''), (A.2), (A.4) and (ⅰ).If (ytn) increasingly con-verges to yt with then (yt) is a g-supersolution, i.e., there ex-ist a (zt)∈Lp(0, T; P; R) and an RCLL increasing process (At) with A0=0 and E[(AT)p] <∞, such that the pair (yt, zt) is the unique solution of the BSDE where (zt) is the weak limit of (ztn) in Lp(0,T; P; R) and for each t, At is the weak limit of Atn in Lp(Ω,Ft,P).Theorem 1.3.3. Assume (Yt) be a right-continuous Lp g-supermartingale on [0,T] with then (Yt) is a g-supersolution on [0,T]: there exists an RCLL increasing process (At) with A0=0 and E[(AT)p] <∞, such that (Yt) coincides with the unique solution (yt) of the BSDETheorem 1.3.4 (Nonlinear Doob-Meyer decomposition theorem). Assume g is independent of y, let (Xt) be a right-continuous Lp g-supermartingale on [0, T] with Then (Xt) has the following decompositionHere (Mt) is a Lp g-martingale and (At) is an RCLL increasing process with A0=0 and E[{AT)p]<∞.Chapter 2: We prove the existence and uniqueness result for Lp(1 2, there exists a positive constant KT not depending on n such that for all n∈N,Theorem 3.2.1 (Central limit theorem for capacities). be a sequence of IID random variables under E. We further assume that E[X1] = E[-X1] = 0. Denote(1) if y is a point at which V is continuous, we have(2) if y is a point at whichνis continuous, we have whereTheorem 3.3.1 (Law of the iterated logarithm for capacities). Let {Xn}n=1∞be a sequence of bounded IID random variables under sublinear expectation E with zero means and bounded variances, i.e.,(Ⅰ)(Ⅱ)(Ⅲ) Suppose that C({xn}) is the cluster set of a sequence of {xn} in R, thenTheorem 3.4.1 (General law of large numbers for sublinear expectations). Let a sequence {Xi}i=1∞which is in a sublinear expectation space (Ω,H,E) satisfy the following conditions:(ⅰ) each Xi+1 is independent of (X1,…, Xi), for i =1,2,(ⅱ) E[Xi] =μi-E[-Xi] =μi, where -∞0,Theorem 3.5.1 (Cramer's upper bound). Let a random sequence {Xn;n≥1} be identically distributed under E[·]. We also assume that each Xn+1 is independent of (X1,…,Xn) for n = 1,2,…under E[?]. Denote Then we have: For any closed set F C R,whereΛ*(·) is a convex rate function.Chapter 4: We obtain some limit results for G-Brownian motion such as the modulus of continuity theorem for G-Brownian motion and How big are the increments of G-Brownian motion.Theorem 4.1.1 (Modulus of continuity theorem for G-Brownian motion). Suppose that (Bt)t≥0 be a 1-dimensional G-Brownian motion with E[B12]=σ2,-E[-B12]=σ2. Then(Ⅰ) Theorem 4.2.1.Suppose that(Bt)t≥0 be a 1-dimensional G-Brownian motion with E[B12]=σ2,-E[-B12]=σ2.Let aT(t≥0) be a nondecreasing function of T for which(ⅰ) 0