This paper is devoted to the construction of a solution for the "Inhomogenous skew Brownian motion" equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied byÉtoré and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.

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In the first part of this paper, we study RBSDEs in the case where the filtration is non-quasi-left-continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools from general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part, we introduce an optimal stopping problem indexed by predictable stopping times with the nonlinear predictable [Formula: see text] expectation induced by an appropriate backward stochastic differential equation (BSDE). We establish some useful properties of [Formula: see text]-supremartingales. Moreover, we show the existence of an optimal predictable stopping time, and we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.

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Given a controlled diffusion with jumps, it is shown under some conditions that there exists a Markov controlled diffusion which has the same cost under any criterion which only depends on the one-dimensional distributions. This result is then used to prove that the finite dimensional marginal distribution of a controlled diffusion with jumps at a prescribed set of time instants can also be attained by using a control from a much smaller class of controls called "nearly Markov controls", extending the work of Borkar [4] to the discontinuous setting.