| By Pardoux-Peng (1990) we know that there exists a unique adapted and square integrable solution to the following backward stochastic differential equation (BSDE inshort)providing that the function g satifies the Lipschitz assumption (A1) and the square integrable assumption (A2). g is said to be the generator of the BSDE (1), (g,T,ξ) is said to be the standard parameters of the BSDE (1). We denote the unique adapted and square integrable solution of the BSDE with standard parameters (g,T, ξ) byIf g also satisfies g(t, y, 0) =0 for any (t, y), then, Y——0(g, T, ξ), denoted by ε_g[ξ], is called g-expectation of ξ; Y_t(g,T,ξ), denoted by ε_g[ξ|F_t], is called conditional g-expectation ofThe original motivation for studying g-expectation comes from the theory of expected utility, which is the foundation of modern mathematical economics. The notion of g-expectation can be considered as a nonlinear extension of the well-known Girsanov transformations.This doctoral thesis study some fundamental problems in g-expectations theory and their applications to risk measure and nonlinear price system.(I) In Chapter 1, we establish a general Representation theorem for generators of BSDEs.This Representation theorem will play a key role when we study some fundamental properties of g-expectations in the following Chapters.For studying a kind of converse comparison problem, Briand et al.(2000) established the following representation theorem:For V (i,y,z) G [0,T[xR x Rd, the following equalityg {t, y, z) = L2- lim - [Yt (g, t + e, y + z ■(Bt+￡ - Bt)) - y] (2)holds for g under two additional conditions that (■y(f, j/,2) is continuous. TTien i/ie following two conditions are equivalent:(i) For each stopping time r g2(t,z);(ii) for each ￡ G L2 (Q,/"T,P), we have: ￡gi [(] > ￡g2 [(].Theorem 2.4.4. (Converse comparison theorem for BSDEs) Let (Al) and (A2) hold for two generators gi and g^- Then the following statements are equivalent: (i) dP x dt - a.s., V(y,2)€RxRrf, ffl(f,y,z) > g2(t,y,z). (ii) For V t e [0, T], ￡ G L2 (0, Tu P): P- a.s.,(Ill) In Chapter 3, we study Jensen's inequality for g-expectations. under the usual assumptions (Al) and (A3), we obtain a necessary and sufficient condition for Jensen's inequality for ^-expectation.We all know that Jensen's inequality is a fundamental inequality in probability theory and martingale theory. Since g-expectation is a kind of nonlinear expectation, it is interesting to study Jensen's inequality for ^-expectation. Roughly speaking, the problem of Jensen's inequality for ^-expectation is:Given a generator g satisfying (Al) and (A3) and a convex function a￡g[Z} + b; (iii) Jensen's inequality for g-expectation holds in general, i.e., for each ￡ G L2 (Q,!Ft,P) and convex function tp : R -? R , if <￡ (￡) 6 I2 (H, Tj, P), thenP - a.s., \ft e [0. T], ￡g [ ip [￡g{Z\Ft)].Corollary 3.2.4. Let g be independent of y, g(t,O) = 0 and be super homogeneousin z. Let (Xtjj^cm ^e a g-martingale and ip : R —> R be a convex function such that ~p{Xt) e L2(Q.,Jrt,P), VY e [O.Tj. Then ($'(^rt))te[o,T] ** a g-submartingale.Theorem 3.2.5. Let g be independent of y; let (Al) and {A3) hold for g. Suppose g is convex in z. Then the following two statements are equivalent:(i) g is positively homogeneous ;(ii) Jensen's inequality for g-expectation holds in general.Theorem 3.3.1. Let (Al) and (.43) hold for g.(i) g is independent of y and dP x dt — a.s.,Vo>0.;6R(l, g{t,az)>ag{t,z).(ii) For each a > 0, b e R, ( t L2 (fi, TT, P), ￡ffK + h) > a￡gls) + &;(iii) Jensen's inequality for g-expectation holds for increasing convex functions.Theorem 3.3.4. Let g satisfy {A\) and (A3) be independent of y. Then the following two statements are equivalent:(i ) dP x dt -a.s., a < 0,z ￡ Rd, g{t,az) > ag{t,z);(ii) Jensen's inequality for g-expectation holds for decreasing convex functions.(IX) In Chapter 4, We apply some results we obtain in Chapter 1 and Chapter 2 to price system and risk measure.Definition 4.2.8. Let g satisfy (Al) and (A3). For VlG L2 (Q, TT,P), we definepZ(X) := ￡g[-X], pf(X) := ￡g[-X\Ft], V i € [0, T).Then p3 is called the static risk measure induced by the g-expectation ￡g, (pf) is called the dynamic risk measure induced by the g-expectation ￡g.Theorem 4.2.6. Let g satisfy (Al) and (A3). Then the following three statements are equivalent:(i) g is independent of y and is sublinear with respect to z. lix) p9 is a static coherent risk measure.(iii) (pf )te[o)rj !S a dynamic coherent risk measure.Theorem 4.2.7. Let g satisfy (Al) and (A3). Then the following three statements are equivalent:(i) g is independent of y and is convex with respect to z.(ii) p9 is a static convex risk measure.(iii) (p?)te[o,ri *s fl dynamic convex risk measure.
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