BSDEs with general filtration driven by Lévy processes, and an application in stochastic controllability

In this paper, we introduce a weak version of the strong solution (the adapted solution used in Pardoux and Peng (1990) [2]), i.e., the transposition solution, to the backward stochastic differential equation (BSDE) with general filtration and random jumps, and study the corresponding well-posedness. The main tools that we employ are the Riesz representation theorem and the Banach fixed point theorem, without using the martingale representation theorem. As an application, we give a definition of controllability to the stochastic linear control system in the sense of the transposition solution and provide a Kalman-type rank condition to guarantee this property.