Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

Abstract

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDE’s $$(\partial _{t}+L)u(t,x)+f(t,x,u,\nabla u\sigma )=0,\quad \forall 0\leqslant t\leqslant T,$$ $$u(T,x)=\Phi (x).$$ We solve this system in the framework of Dirichlet spaces and employ the symmetric Markov process of infinitesimal operator L in order to obtain a precised version of the solution u by solving the corresponding system of backward stochastic differential equations. This precised version verifies pointwise the so called “mild equation”, which is equivalent to the above PDE. As a technical ingrediend we prove a representation theorem for arbitrary martingales which generalises a result of Fukushima for martingale additive functionals. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.