BSDE and generalized Dirichlet forms: the finite dimensional case

TL;DR

This paper develops a probabilistic framework using generalized Dirichlet forms to solve a class of degenerate quasi-linear backward PDEs, extending stochastic calculus and martingale representation to this setting.

Contribution

It introduces a novel approach combining generalized Dirichlet forms and stochastic calculus to solve degenerate PDE systems and generalizes the martingale representation theorem.

Findings

Probabilistic representation of solutions via backward SDEs.

Extension of stochastic calculus to degenerate operators.

Generalization of the martingale representation theorem.

Abstract

We consider the following quasi-linear parabolic system of backward partial differential equations: $(\partial_t+L)u+f(\cdot,\cdot,u, \nabla u\sigma)=0$ on $[0,T]\times \mathbb{R}^d\qquad u_T=\phi$, 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 $\nabla u$.