THE STOCHASTIC MAXIMUM PRINCIPLE FOR LINEAR, CONVEX OPTIMAL-CONTROL WITH RANDOM-COEFFICIENTS

This paper considers a stochastic control problem with linear dynamics, convex cost criterion, and convex state constraint, in which the control enters both the drift and diffusion coefficients. These coefficients are allowed to be random, and no L(p)-bounds are imposed on the control. An explicit solution for the adjoint equation and a global stochastic maximum principle are obtained for this model. This is evidently the first version of the stochastic maximum principle that covers the consumption-investment problem. The mathematical tools are those of stochastic calculus and convex analysis. When it is assumed, as in other versions of the stochastic maximum principle, that the admissible controls are square-integrable, not only a necessary but also a sufficient condition for optimality is obtained. It is then shown that this particular case of the general model may be applied to solve a variety of problems in stochastic control, including the linear-regulator, predicted-miss, and Benes problems.