Linear Quadratic Optimal Stochastic Control with Random Coefficients

The purpose of this paper is to apply the methods developed in [1] and [2] to solve the problem of optimal stochastic control for a linear quadratic system. After proving some preliminary existence results on stochastic differential equations, we show the existence of an optimal control. The introduction of an ad joint variable enables us to derive extremality conditions: the control is thus obtained in random “feedback” form. By using a method close to the one used by Lions in [4] for the control of partial differential equations, a priori majorations are obtained. A formal Riccati equation is then written down, and the existence of its solution is proved under rather general assumptions. For a more detailed treatment of some examples, the reader is referred to [1].