Necessary and sufficient conditions for the optimal control of systems with random coefficients

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. The authors obtain for this model an explicit solution for the adjoint equation, and a global stochastic maximum principle. This is the first version of the stochastic maximum principle that covers the consumption-investment problem. When the authors assume, as in other versions of the stochastic maximum principle, that the admissible controls are square-integrable, they obtain not only a necessary but also a sufficient condition for optimality. The mathematical tools are those of stochastic calculus and convex analysis