Euclidean Space And Function Space In The Backward Stochastic Equations

The thesis is concerned with backward stochastic differential equations (BSDEs, for short), backward doubly stochastic differential equations (BDSDEs, for short), and semi-linear systems of backward stochastic integral partial differential equations driven by both Brownian motions and Poisson point processes. It consists of two parts.The first part is concerned with the Lp solutions (1< p< 2) to BSDEs and BDSDEs. In Chapter 2, via approximating the generator by smooth functions, we first show that there is unique L2 solution to a one-dimensional BSDE if the generator f is independent of the first unknown variable y, uniformly continuous with respect to the second unknown variable z, and of linear growth. Then, by approximating the generator by monotone functions, we prove that there is unique Lp solution to a one-dimensional BSDE if the generator f is uniformly continuous with respect to the unknown variables (y,z) and is of linear growth, and if the modulus of continuity of f with respect to the first unknown variable y is of certain integrability. In Chapter 3, by means of weak convergence, we first prove that there is unique L2 solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a rather general growth in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than 1. Then we prove an Ito formula which involves both forward Ito integral and backward Ito integral. Finally, by establishing some a prior estimates, we show that there is unique Lp solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a more general integrability in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than (?)The second part is concerned with, from a probabilistic point of view, semi-linear systems of backward stochastic integral partial differential equations, which arise from the study of non-Markovian forward-backward stochastic differential equations driven by both Brownian motions and Poisson point processes. In Chap-ter 4, we first establish an Ito-Wentzell formula for the semimartingales driven by both Brownian motions and Poisson point processes. Then we obtain the existence and uniqueness of solutions to non-degenerate stochastic evolution equations via Galerkin approximation and the contraction mapping principle. Applying the ab-stract result to concrete equations and using the skill of approximation, we further prove the existence and uniqueness of solutions to degenerate stochastic integral partial differential equations. Combining the Ito-Wentzell formula, we derive the stochastic integral partial differential equation for the inverse of a stochastic flow generated by a stochastic differential equation driven by both Brownian motion and Poisson point process. Then we analyze the regularity of the solutions to the BSDEs with Poisson jumps. Finally, by the composition of the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow stated above, we construct the classical solution to the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.