Generalized Solutions in the Optimal Control of Diffusions

Abstract

We consider optimal control of Markov diffusion procesess n-dimensional R ⁿ, on a finite time interval t ≤ s ≤ T. The dynamics of the process x _s being controlled are governed by the stochastic differential equation

$$d{x_s} = b(s,x,{u_s})ds + \sigma (s,{x_s},{u_s})d{\omega _s} $$

(1)

with initial data x _s = x. The control process u _s takes values in a control space Y, and is progressively measurable with respect to the filtration of the brownian motion process w _s . The objective is to minimize an expected cost

$$ {J^U}(t,x) = {E_{tx}}\smallint _t^t{\ell _o}(s,{x_s}{u_s})ds $$

(1.2)

.

This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the NSF and Office of Naval Research.

Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912. Partially supported by NSF under grant No. MCS 8121940, by ONR under contract No. N0014-83K-0542 and by AFOSR under contract No. 91–0116–0

Department of Mathematics, University of Washington, Seattle, Washington 98195. Part of this research was done while the second author was visiting Brown University.