INDEFINITE STOCHASTIC LINEAR-QUADRATIC OPTIMAL CONTROL PROBLEMS WITH RANDOM COEFFICIENTS: CLOSED-LOOP REPRESENTATION OF OPEN-LOOP OPTIMAL CONTROLS

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly more general than the classical conditions that the weighting matrix-valued processes are positive (semi-) definite.

Automated summary

This paper explores a stochastic linear-quadratic optimal control problem in a finite time horizon, showing that convexity of the cost functional is necessary for the existence of an open-loop optimal control, while uniform convexity is sufficient. It also provides sufficient conditions for uniform convexity of the cost functional, which are more general than classical conditions.