Duality and the well-posedness of a martingale problem

Andrej Depperschmidt, Andreas Greven, Peter Pfaffelhuber

Published 2019-04-02Version 1

For two Polish state spaces $E_X$ and $E_Y$, and an operator $G_X$, we obtain existence and uniqueness of a $G_X$-martingale problem provided there is a dual process $Y$ on $E_Y$ solving a $G_Y$-martingale problem. Duality here means the existence of a rich function $H$ and transition kernels $(\mu_t)_{t\geq 0}$ on $E_X$ such that $\mathbf E_y[H(x,Y_t)] = \int \mu_t(x,dx') H(x',y)$ for all $(x,y) \in E_X \times E_Y$ and $t\geq 0$. While duality is well-known to imply uniqueness of the $G_X$-martingale problem, we give here a set of conditions under which duality also implies existence without using approximation techniques. As examples, we treat branching and resampling models, such as Feller's branching diffusion and the Fleming-Viot superprocess.