Convex Stochastic Control and Conjugate Duality in a Problem of Unconstrained Utility Maximization Under a Regime Switching Model
Situ, Aaron Xin
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In this thesis, we examine a problem of convex stochastic optimal control applied to mathematical finance. The goal is to maximize the expected utility from wealth at close of trade (or terminal wealth) under a regime switching model. The presence of regime switching constitutes a definite challenge, and in order to keep the analysis tractable we therefore adopt a market model which is in other respects quite simple, and in particular does not involve margin payments, inter-temporal consumption or portfolio constraints. The asset prices will be modeled by classical Ito processes, and the market parameters will be dependent on the underlying Brownian Motion as well as a finite-state Markov Chain which represents the "regime switching" aspect of the market model. We use conjugate duality to construct a dual optimization problem and establish optimality relations between (putative) solutions of the dual and primal problems. We then apply these optimality relations to two specific types of utility functions, namely the power utility and logarithmic utility functions, and for these utility functions we obtain the optimal portfolios in completely explicit and implementable form.