We will discuss two new regularity properties of the optimal stopping problem for Levy processes with a non-degenerate diffusion component. First, under some conditions on the Levy measure, we will show that the value function of the optimal stopping problem is a classical solution of the variational inequality. This result has been known for Levy processes of finite activity. We extend it to the general Levy processes of infinite activity. Second, we will discuss a specific example of the optimal stopping problem: the American option pricing problem. We will show that the optimal exercise boundary / free boundary of the American put in jump diffusion models is continuously differentiable (except at the maturity). Moreover, the boundary is shown to be infinitely differentiable under a regularity assumption on the jump distribution. These are joint works with Prof. Erhan Bayraktar. ( Slides)
Speaker: Hao Xing, University of Michigan