Dynamic Convex Risk Measures And Related Problems

The famous Allais paradox and Ellsberg paradox motivate scientists to pay attention to considering economic and financial problems under the framework of nonlinear expec-tations. In 1997, Peng introduced the concept of g-expectations via nonlinear backward stochastic differential equations and showed that g-expectations are filtration-consistent nonlinear expectations. This thesis aims at studying the representations of dynamic con-vex risk measures and related problems based on the theory of g-expectations.Chapter 1 of this thesis briefly introduces the research background and our main re-sults. Starting from Chapter 2, the representations of dynamic convex risk measures and related problems based on the theory of g-expectations are investigated in depth and some significant advances are obtained.Chapter 2 presents the relationship of domination between linear expectations and convex expectations. We investigate the minimal members of convex expectations with constraints. By using a constructed method, we overcome the deficiencies in the proofs of the main theorems in Huang-Jia (2011) and obtain a series of interesting results (see Theorems 2.1-2.4 and Theorem 2.6). We also build the relationship between the famous Hahn-Banach theorem and minimal members of the set of convex expectations with single or two domination conditions under the framework of nonlinear expectations in Coquet-Hu-Memin-Peng (2002) (see Theorem 2.5). Furthermore, we study the minimal members of convex g-expectations. We establish the sandwich theorem for g-expectations and prove that the minimal members of convex g-expectations are linear g-expectations (see Theorem 2.7 and Theorem 2.8). Besides, as for the minimal penalty function of condition convex risk measure induced by a g-expectation, we obtain a sufficient and necessary condition for probability measures with zero penalty (see Proposition 2.4).Chapter 3 considers the representations of time consistent dynamic convex risk mea-sures induced by g-expectations. We show the representation term of dynamic convex risk measure induced by a g-expectation in the view of generator g, with the help of cocycle property of the minimal penalty function and the family of probability measures controlled by some sublinear g-expectation (see Theorems 3.3-3.6). We also study the penalty term of dynamic convex risk measures induced by g-expectations and obtain a sufficient and necessary condition for the fact that penalty function in Barrieu-El Karoui (2005) is the minimal one (see Theorem 3.7). The research of this chapter is directly motivated by the results in Jiang (2008) and our results of this chapter improve some corresponding results in Rosazza Gianin (2006), Barrieu-El Karoui (2005), Chen-Kulperger (2006) and Jiang (2009).Chapter 4 investigates the representations of time consistent dynamic convex risk measures based on the theory of g-expectations. Delbaen-Peng-Rosazza Gianin (2010) considered the representations of a kind of time consist dynamic concave utilities by ap-plying the theory of g-expectations with two additional assumptions. We propose a new assumption to replace the additional assumptions in Delbaen-Peng-Rosazza Gianin (2010). Therefore our result (see Theorems 4.1) improves the result in Delbaen-Peng-Rosazza Gi-anin (2010). Besides, the result in this chapter also extends the results in Chapter 3 in the view of nonlinear expectations.Chapter 5 discusses the dynamic convex risk measures for processes induced by back-ward stochastic differential equations. Under some assumptions on generator g, where g need not be independent of y, Penner-Revaillar (2015) showed that the risk measures induced by some kinds of backward stochastic differential equations are time consistent dynamic convex risk measures for processes. This chapter proves that the risk measures induced by the backward stochastic differential equations are time consistent dynamic con-vex risk measures for processes if and only if generator g satisfies the assumptions listed in Penner-Revaillar (2015) (see Theorem 5.5). The research of this chapter is motivat-ed by the one-to-one correspondence between g-expectations and risk measures induced by g-expectations in Jiang (2008) and our results generalize the corresponding results in Penner-Revaillar (2015).