## Numerical Approximations of Stochastic Optimal Stopping and Control Problems

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2007##### Author

Siska, David

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##### Abstract

We study numerical approximations for the payoff function of the stochastic optimal stopping
and control problem. It is known that the payoff function of the optimal stopping and control
problem corresponds to the solution of a normalized Bellman PDE.
The principal aim of this thesis is to study the rate at which finite difference approximations,
derived from the normalized Bellman PDE, converge to the payoff function of the optimal
stopping and control problem. We do this by extending results of N.V. Krylov from the Bellman
equation to the normalized Bellman equation. To our best knowledge, until recently, no
results about the rate of convergence of finite difference approximations to Bellman equations
have been known. A major breakthrough has been made by N. V. Krylov. He proved rate of
rate of convergence of tau 1/4 + h 1/2 where tau and h are the step sizes in time and space respectively.
We will use the known idea of randomized stopping to give a direct proof showing that
optimal stopping and control problems can be rewritten as pure optimal control problems by
introducing a new control parameter and by allowing the reward and discounting functions to
be unbounded in the control parameter.
We extend important results of N. V. Krylov on the numerical solutions to the Bellman
equations to the normalized Bellman equations associated with the optimal stopping of controlled
diffusion processes. We obtain the same rate of convergence of tau1/4 + h1/2. This rate of
convergence holds for finite difference schemes defined on a grid on the whole space [0, T]×Rd
i.e. on a grid with infinitely many elements. This leads to the study of localization error, which
arises when restricting the finite difference approximations to a cylindrical domain.
As an application of our results, we consider an optimal stopping problem from mathematical
finance: the pricing of American put option on multiple assets. We prove the rate of
convergence of tau1/4 + h1/2 for the finite difference approximations.