Duality and the well-posedness of a martingale problem

Andrej Depperschmidt, Andreas Greven and Peter Pfaffelhuber

Theoretical Population Biology, 2024, vol. 159, issue C, 59-73

Abstract: 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.

Keywords: Duality; Martingale problems; Construction of solutions of martingale problems; Transition semigroups; Fleming–Viot process; Cannings process; Branching process; Spatial population models (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:eee:thpobi:v:159:y:2024:i:c:p:59-73

DOI: 10.1016/j.tpb.2024.07.003

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