New Results On Backward Stochastic Equations And Their Applications

This dissertation is concerned with backward stochastic differential equations (BSDEs for short) and g-expectations with jumps, time-consistent dynamic convex and coherent risk mea-sures, and the Cauchy problem of super-parabolic linear backward stochastic partial differential equations (BSPDEs) in Holder spaces. It is organized as follows:In the first chapter, we first recall some results on BSDEs with jumps. Then we introduce the representation of the generator of BSDEs with jumps by the estimates of SDEs and BSDEs and a forward-backward SDEs. At last, we prove the converse comparison theorem of BSDEs with jumps.In the second chapter, under the filtration generated by a Brownian motion and a Poisson random measure, the relation is discussed between g-expectations and time-consistent dynamic convex and coherent risk measures using the results in the first chapter. Also the integral rep-resentation is discussed for the minimal penalty term of a time-consistent dynamic convex risk measure.The third chapter is concerned with the Cauchy problem of super-parabolic linear back-ward stochastic partial differential equations in Holder spaces. Some Holder spaces of Ba-nach space valued functionals are defined. In these Holder spaces, we first consider the linear BSPDEs with space-invariant coefficients. By the potential of heat equation, we discuss the existence and uniqueness of the solution, and its well-posedness. It is shown that when the terminal condition is C1+α-Holder continuous and free term is Cα, there is a unique pair of solution which is C2+α x Cα-Holder continuous. Then, by methods of freezing coefficients and continuity, we solve BSPDEs with variable coefficients.