Forward-backward SDE games and stochastic control under model uncertainty

Abstract

We study optimal stochastic control problems under model uncertainty. We rewrite such problems as (zero-sum) stochastic di erential games of forward-backward stochastic di erential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case ( nding conditions for saddle points) and for the non-zero sum games ( nding conditions for Nash equilibria). We then apply these results to study optimal portfolio and consumption problems under model uncertainty. We combine the optimality conditions given by the stochastic maximum principles with Malliavin calculus to obtain a set of equations which determine the optimal strategies.