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Utility Maximization in a Regime Switching Model with Convex Portfolio Constraints and Margin Requirements: Optimality Relations and Explicit Solutions

We study a problem of stochastic control in mathematical finance, with the goal of maximizing expected utility of investment and consumption over a finite trading horizon. The asset prices are modeled by Ito processes, for which the market parameters are subject to regime switching in the sense of being adapted to the joint filtration of the driving Brownian motion and a finite-state Markov chain which models “regime states” of the market. The vector of portfolios is constrained to a specified closed and convex set, and margin payments are levied on the investor, resulting in a wealth equation which is nonlinear in the portfolio. We proceed by the method of conjugate duality to construct a dual optimization problem together with optimality relations between putative solutions of the given (i.e., “primal'') optimization problem and the dual optimization problem. These optimality relations are then used to address the specific cases of power-type and logarithmic utility functions, with convex cone portfolio...