TALK by K. Ross

April 3, 2008

Optimal stopping and free boundary characterizations for some Brownian control problems

Kevin Ross (Stanford)

We study a singular stochastic control problem with state constraints in two dimensions. We show that the value function is continuously differentiable and its directional derivatives are the value functions of certain optimal stopping problems. Guided by the optimal stopping problem we then introduce the associated no-action region and the free boundary and show that, under appropriate conditions, an optimally controlled process is a Brownian motion in the no-action region with reflection at the free boundary. This proves a conjecture of Martins, Shreve and Soner (SIAM J. Control Optim., 34: 2133-2171, 1996) on the form of an optimal control for this class of singular control problems. An important issue in our analysis is that the running cost is Lipschitz but not differentiable. This lack of smoothness is one of the key obstacles in establishing regularity of the free boundary and of the value function. We show that the free boundary is Lipschitz and that the value function is C2 in the interior of the no-action region. We then use a verification argument applied to a suitable C2 approximation of the value function to establish optimality of the conjectured control.

This is a joint work with Amarjit Budhiraja.