Duality and the well-posedness of a martingale problem

For two Polish state spaces EX and EY, and an operator GX, we obtain existence and uniqueness of a GX-martingale problem provided there is a bounded continuous duality function H on EX×EY together with a dual process Y on EY which is the unique solution of a GY-martingale problem. For the corresponding solutions (Xt)t≥0 and (Yt)t≥0, duality with respect to a function H in its simplest form means that the relation Ex[H(Xt,y)]=Ey[H(x,Yt)] holds for all (x,y)∈EX×EY and t≥0. While duality is well-known to imply uniqueness of the GX-martingale problem, we give here a set of conditions under which duality also implies existence without using approximating sequences of processes of a different kind (e.g. jump processes to approximate diffusions) which is a widespread strategy for proving existence of solutions of martingale problems. Given the process (Yt)t≥0 and a duality function H, to prove existence of (Xt)t≥0 one has to show that the r.h.s. of the duality relation defines for each y a measure on EX, i.e. there are transition kernels (μt)t≥0 from EX to EX such that Ey[H(x,Yt)]=∫μt(x,dx′)H(x′,y) for all (x,y)∈EX×EY and all t≥0.