The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls

This paper is concerned with a Pontryagin’s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls. This kind of control problems is motivated by some interesting phenomena arising from economics and finance. We establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis. To explain the theoretical results, we apply them to a production and consumption choice problem.