Convex Stochastic Control and Conjugate Duality in a Problem of Unconstrained Utility Maximization Under a Regime Switching Model
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Date
2015-05-22
Authors
Situ, Aaron Xin
Journal Title
Journal ISSN
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Publisher
University of Waterloo
Abstract
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.
Description
Keywords
utility optimization, regime switching model, conjugate duality, stochastic calculus, convex stochastic control, mathematical finance