A Degree Method for Free Boundaries in Stochastic Control

Abstract

In stochastic control problems with a bounded control set, the Bellman-Hamilton Jacobi equation leads to two-sided free-boundary problems for the switching surfaces, expressible as an equivalent set of integral equations containing the boundary functions in a very implicit way that seems to preclude the standard method used in the Stefan problem. It is natural then to try to use the topological Leray-Schauder methods to study the properties of solutions. We apply such an approach to the sample problem:$\min _u E\int_0^T {[f(x_t ) + | {u(x_t ,t)} |]} dt$, subject to $dx_t = u(x_t ,t)dt + dw_t $, $| u | \leqq 1$, with $w_t $ a Wiener process. The absolute value cost $| u |$ leads to finding the boundaries of a “dead zone” in $(x,t)$-space that separates the zones $u = \pm 1$ for the optimal u . The a priori bounds requisite for the Leray-Schauder approach come from usual probabilistic and PDE estimates. Then the integral equations are shown to have the form (homeomorphism $ + $ compact) for which a degree theory is available. Finally, a simple homotopy shows that the free boundary is continuously differentiable; separate arguments establish its uniqueness and monotonicity.