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Abstract
Optimizations play an increasingly indispensable role in financial decisions and financialmodels. Many problems in mathematical finance, such as asset allocation, trading strategy,and derivative pricing, are now routinely and efficiently approached using optimization. Notuntil recently have stochastic approximation methods been applied to solve optimizationproblems in finance.This dissertation is concerned with stochastic approximation algorithms and their appli-cations in financial optimization problems. The first part of this dissertation concerns tradinga mean-reverting asset. The strategy is to determine a low price to buy and a high price to sellso that the expected return is maximized. Slippage cost is imposed on each transaction. Oureffort is devoted to developing a recursive stochastic approximation type algorithm to esti-mate the desired selling and buying prices. In the second part of this dissertation we considerthe trailing stop strategy. Trailing stops are often used in stock trading to limit the maximumof a possible loss and to lock in a profit. We develop stochastic approximation algorithmsto estimate the optimal trailing stop percentage. A modification using projection is devel-oped to ensure that the approximation sequence constructed stays in a reasonable range.In both parts, we also study the convergence and the rate of convergence. Simulations andreal market data are used to demonstrate the performance of the proposed algorithms. Theadvantage of using stochastic approximation in stock trading is that the underlying asset ismodel free. Only observed stock prices are required, so it can be performed on line to provideguidelines for stock trading. Other than in stock trading, stochastic approximation methodscan also be used in parameter estimations. In the last part of this dissertation, we consider aregime switching option pricing model. The underlying stock price evolves according to twogeometric Brownian motions coupled by a continuous-time finite state Markov chain. Recur-sive stochastic approximation algorithms are developed to estimate the implied volatility.Convergence of the algorithm is obtained and the rate of convergence is also ascertained.Then real market data are used to compare our algorithms with other schemes.