dc.contributor.advisor Gyongy, Istvan en dc.contributor.author Siska, David en dc.date.accessioned 2008-11-27T13:30:32Z dc.date.available 2008-11-27T13:30:32Z dc.date.issued 2007 dc.identifier.uri http://hdl.handle.net/1842/2571 dc.description.abstract We study numerical approximations for the payoff function of the stochastic optimal stopping en 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. dc.format.extent 624658 bytes en dc.format.mimetype application/pdf en dc.language.iso en dc.subject Mathematics en dc.subject Stochatic control en dc.subject Bellman PDE en dc.title Numerical Approximations of Stochastic Optimal Stopping and Control Problems en dc.type Thesis or Dissertation en dc.type.qualificationlevel Doctoral en dc.type.qualificationname PhD Doctor of Philosophy en
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