Regression and convex switching system methods for stochastic control problems with applications to multiple-exercise options

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In this thesis, we develop several new simulation-based algorithms for solving some important classes of optimal stochastic control problems. In particular, these methods are aimed at providing good approximate solutions to problems that involve a high-dimensional underlying processes. These algorithms are of the primal-dual kind and therefore provide a gauge of the distance to optimality of the given approximate solutions to the optimal one. These methods will be used in the pricing of the multiple-exercise option. In Chapter 1, we conduct a review of the literature that is relevant to the pricing of the multiple-exercise option and the primal and dual methods that we will be developing in this thesis. In the next two chapters of the thesis, we focus on regression-based dual methods for optimal multiple stopping problems in probability theory. In particular, we concentrate on finding upper bounds on the price of the multiple-exercise option as it sits within this framework. In Chapter 2, we derive an additive dual for the multiple-exercise options using financial arguments, and see that this approach leads to the construction of an algorithm that has greater computational efficiency than other methods in the literature. In Chapter 3, we derive the first known dual of the multiplicative kind for the multiple-exercise option and devise a tractable algorithm to compute it. In the penultimate chapter of the thesis, we focus on a new class of algorithms that are based on what is known as convex switching system. These algorithms provide approximate solutions to the more general class of optimal stochastic switching problems. In Chapter 4, techniques based on combinations of rigorous theory and heuristics arguments are used to improve the efficiency and applicability of the method. We then devise algorithms of the primal-dual kind to assess the accuracy of this approach. Chapter 5 concludes.
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