Properties For Nonlinear Expectations And L~ρ Solutions Of Backward Stochastic Differential Equations

It is well-known that the theory of nonlinear expectations has been exten-sively studied. Motivated by the theory of expected utility, Peng introduced the notion of g-expectation via a nonlinear backward stochastic differential equation (BSDE) in [50]. He proved that the g-expectation preserves many of properties of the classical mathematical expectation, but not the linearity property, and thus the g-expectation is a type of nonlinear mathematical expectation. Motivated by the risk measures and stochastic volatility problems in finance, in2006Peng intro-duce a new kind of nonlinear expectation-G-expectation by a nonlinear parabolic partial differential equation with an infinitesimel generetor G in [51]. Due to the impertantment applications of g-expectation and G-expectation in many fields, a considerable amount of works have been devoted to studying g-expectation and G-expectation (sec Chen et al.([15],[16]), Hu ([31]), Jiang et al.([36]), Jia et al.([33]), Peng ([54],[55]),Wang ([71],[72]), Chen ([12]), Hu ([28]), Hu and Zhang ([29]), Hu and Peng ([30]), Jiang and Chen ([36]), Chen ([10]), Bai and Buckdahn ([2]), etc). In recent two decades, the theory of backward stochastic differential equation (BSDE) is a very hot question. This theory can be traced back to Bismut ([3]) who studied the linear case. In1990, Pardoux and Peng ([47]) proved the wellposcdncss for nonlinear BSDEs. Since then, backward stochastic differential equations have gradually become an important mathematical tool in many fields such as financial mathematics, stochastic control, partial differential equations, stochastic games, etc. Since the pioneering paper of Pardoux and Peng many researchers have been working on this subject. One can sec details in Briand ([4]), Briand ct al.([5],[6]), Briand and Hu ([7],[8]) Chen and Wang ([13]), Coquet et al.([18]), El Karoui and Peng ([20]), Fan and Jiang ([22]), Jia ([32]), Jiang ([34],[35]), Lcpelticr and Martin ([39]), Mao ([43]), Tian et al.([60]), Wang and Huang ([65]), Wang et al.([66]), etc.This thesis consists of four chapters. In the following, we list the main results of this thesis.In Chapter1, one side, we study the Jensen's inequality for non-linear expectations: Jensen's inequality for sublinear expectations and Jensen's inequality for minimal mathematical expectations. Another side, we study the property of capacity induced by sublinear expecta-tions: Borel-Cantelli Lemma for capacity.Now let us give the main results. Firstly, we get the Jensen's inequality for sublinear expectations.Theorem1.3.3Let (Ω,H,E) is a sublinear expectation space, if h is a continuous non-decreasing concave function defined on R. For(?)X∈H,if h(X)∈H, then the following Jensen's inequality holds E[h(X)]≤h(E[X]). Theorem1.3.5Let (Ω,H, E) is a sublinear expectation space, if g is a continuous concvex function defined on R. For (?)X∈H, if g(X)∈H, then the following Jensen's inequality holds g(E[X])≤E[g(X)].For the minimal mathematical expectations E, we get the Jensen's inequality for E.Theorem1.3.9If h is a continuous concave function defined on R. For all integrable random variable X, if h(X) is integrable, then the following Jensen's inequality holds E[h(X)]<h(E[X]).Theorem1.3.10If g is a continuous non-decreasing convex function defined on R. For all integrable random variable X, if g(X) is integrable, then the following Jensen's inequality holds g(E[X])<E[g(X)].At last of this paper, we give two examples to check that the Jensen's inequal-ity in Theorem1.3.3is not true for continuous, non-increasing concave functions and the Jensen's inequality in Theorem1.3.10is not true for continuous, decreas-ing convex functions.The main results about Borel-Cantelli Lemma for capacity:Theorem1.4.2Let{Ai,i≥1} be a sequence of events in F,(V,v) be a pair of capacities induced by sublinear expectation E. (Ⅱ) There exist two constants C and K≥1, for all i,j>C and i≠j, such thatBorel-Cantelli Lemma for capacity under a more general condition.Theorem1.4.4Let{Ai,i≥1} be a sequence of events in F,(V,v) be a pair of capacities induced by sublinear expectation E.(Ⅱ) Let H be an arbitrary real constant, setIn Chapter2, we study the LP solutions of backward stochastic differential equations. We obtain the representation theorem for gen-erator g of BSDEs in Lp spaces and Hp1, spaces of processes; the existence and uniqueness theorem of Lp solutions for a class of BSDEs with non-uniformly Lipschitz coefficients.Now we give the main results: Theorem2.3.3Let1≤p'≤p;(A1) and (A2) hold for g;(H1)-(H3) hold for b and σ. Then for each (x,y,q)∈Rn×R×Rn, dt-a.s. in [0, T[, the following two statements are equivalent: (1)Lp'-g(t,y, σ*(t, x)q)+...