Stochastic Maximum Principle with Default

Abstract We study stochastic control problems in a financial market subject to default risk, modeled via a progressive enlargement of filtration. The default event is represented by a random time, and the associated information flow is incorporated through an enlarged filtration framework. We establish a stochastic maximum principle for controlled diffusion processes with a single jump at the default time. The approach relies on the martingale representation under the enlarged filtration, which allows us to derive the adjoint equations and characterize necessary optimality conditions. The presence of default introduces additional terms in both the state dynamics and the adjoint processes, reflecting the impact of jump risk and information flow on optimal strategies. We apply the theoretical results to two explicitly solvable portfolio optimization problems under default risk. In particular, we obtain closed form expressions for the optimal controls in both the logarithmic and exponential utility cases. We further analyze how the default mechanism influences optimal strategies through the filtration structure and model parameters. These results contribute to a deeper understanding of optimal decision making in the presence of random time events.