The aim of this paper is to characterize the Snell envelope of a given P-measurable process l as the minimal solution of some backward stochastic differential equation with lower general reflecting barriers and to prove that this minimal solution exists.
Using the balayage formula, we prove an inequality between the measures associated to local times of semimartingales. Our result extends the comparison theorem of local times of Ouknine $(1988)$, which is useful in the study of stochastic differentia
l equations. The inequality presented in this paper covers the discontinuous case. Moreover, we study the pathwise uniqueness of some stochastic differential equations involving local time of unknown process.
We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a maximal solutio
n for GRBSDE when the terminal condition xi is F_T-measurable, the coefficient f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and the reflecting barriers L and U are just right continuous left limited. The result is proved without assuming any P-integrability conditions.
