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In this paper, we summarize some of our recent results for the American put option. Specifically, we will discuss existence and uniqueness of the solution and regularity of the early exercise boundary. We shall also provide estimates for the near-expiry and far-from-expiry behavior of this boundary as well its convexity/nonconvexity. [ABSTRACT FROM AUTHOR]

It is known that the zero level set of the solution of the Allen-Cahn equation ut- △u =ε-2(u - u3) approaches, as ε→ 0, to the solution of the motion by mean curvature flow. In this paper, we consider the convergence of numerical solutions of a discretized Allen-Cahn equation. In particular,we show that if for some p > 1, the space mesh size △x satisfies △z = O(εp), and the time mesh size satisfies the Courant-Friedrich.