Convergence of a Numerical Scheme for Optimal Stopping over a Finite Time-Horizon of a Diffusion with Singular Behavior

Abstract

This dissertation establishes an approximation scheme for finite time-horizon stopping problems involving a singular stochastic process on a compact state space, characterized by a singular martingale problem.The stopping problem is converted to a linear program (LP) with infinitely many constraints and variables having infinite degrees of freedom. To obtain a numerical solution, the infinite-dimensional LP is converted into a finite LP.The original LP is approximated by a sequence of finite LPs, limiting to both a finite set of constraints and a finite-dimensional solution space. The value of an optimal approximate solution is shown to be arbitrarily close to the optimal value of original LP, and hence of the stopping probem, with increasing refinement of the approximation. Feasibility of the approximate solutions is guaranteed due weak convergence of measures, but only in the limit. The problem of pricing an American floating strike lookback call option can be reformulated to fit the models covered by this dissertation. The price and the stopping boundary can therefore be approximated using this scheme.