Backward stochastic differential equation with local time

In this paper we deal with the following backward stochastic differential equation: where W is a d-dimensional Brownian motion is the symmetric local time of Fat the level a, v is a signed measure on is a -measurable random variable in and is an adapted map from to . If h is continuous with linear growth, we show the existence of a solution (Y,Z) for this backward equation. Some applications of this result, in connection with partial differential equations, and with linear quadratic stochastic control problem, are also given