Convergence Of A Numerical Scheme For Optimal Stopping Of A Diffusion Over A Finite Time-Horizon

Mr. Steffen Domke
Graduate Student
University of Wisconsin-Milwaukee

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.

Advisor:
Richard Stockbridge

Committee Members:
David Spade, Lei Wang, Jeb Willenbring, and Chao Zhu