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 PDE \begin{eqnarray*} (_t+L)u(t,x)+f(t,x,u,u) &=&0,0tT u(T,x) &=&(x) \end{eqna- rray*} and employ the symmetric Markov process of infinitesimal operator L in order to give a probabilistic interpretation for the solution u, i.e. we solve the corresponding BSDE. We obtain also a representation theorem for martingales which represents a generalization of the representation theorem given by Fukushima for additive functional martingales. This permits us to solve general (non-Markov) BSDE's with semi-linear terms. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to u. Finally we prove a comparison theorem and use it in order to solve a stochastic control problem.