Martingale driven BSDEs, PDEs and other related deterministic problems

We focus on a class of BSDEs driven by a cadlag martingale and corresponding Markov type BSDE which arise when the randomness of the driver appears through a Markov process. To those BSDEs we associate a deterministic problem which, when the Markov process is a Brownian diffusion, is nothing else but a parabolic type PDE. The solution of the deterministic problem is intended as decoupled mild solution, and it is formulated with the help of a time-inhomogeneous semigroup.