BSDE and generalized Dirichlet forms: The finite- dimensional case

Zhu R. BSDE and generalized Dirichlet forms: The finite- dimensional case. Infinite Dimensional Analysis Quantum Probability And Related Topics. 2012;15(4): 1250022.We consider the following quasi-linear parabolic system of backward partial differential equations (partial derivative(t) + L)u + f(., ., u, del u sigma) = 0 on [0, T] x R-d u(T) = phi, where L is a possibly degenerate second-order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to del(u)