A stochastic maximum principle with dissipativity conditions

In this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process. We drop the usual Lipschitz assumption on the drift term and substitute it with dissipativity conditions, allowing polynomial growth. The control enters both the drift and the diffusion term and takes values in a general metric space