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

Abstract

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 Itô 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 constraints, and a higher rate of interest for borrowing than for lending. We get completely explicit optimal portfolios and characterize the optimal consumption rate as the solution of a backward stochastic differential equation (BSDE) "driven" by the canonical martingales of the regime-state Markov chain. For the power utility function this is a rather unconventional BSDE, to which standard existence results do not apply, and accordingly we establish existence and uniqueness of solutions for this BSDE. Introduction.We address a problem of stochastic optimal control with the goal of trading over a finite time horizon to maximize expected utility from wealth at close of trade together with utility from intertemporal consumption. Our market model is quite classical, comprising a single risk-free asset and a finite number of risky assets, the prices of which are modeled by continuous-time Itô processes driven by a standard multidimensional Brownian motion. The problem includes (i) portfolio constraints, namely, the portfolio vector of holdings in the risky assets must take values in a specified closed convex set; (ii) margin payments levied on the investor, resulting in a wealth equation which is generally nonlinear in the portfolio; (iii) regime switching in the market model in the following sense: besides the standard Brownian motion driving the stock price models a further source of randomness in the model is a finite-state continuous-time Markov chain which is independent of the Brownian motion, it being stipulated that the market parameters (i.e., the risk-free interest rate, mean rate of return, and volatility of stocks) are adapted to the joint filtration of the Brownian motion and the Markov chain. Each state of the Markov chain models a market "mode" or "regime state"; a simple but apt example is that of a two-state

Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints

“…In the vast majority of the literature, it is often assumed that the investor is able to select his consumption-portfolio strategies with some constraints. For time-additive utilities, on the one hand, one can refer to Cvitanic and Karatzas (1992), Rouge and El Karoui (2000) and Bian, Chen and Xu (2019) for convex constraints, and Hu, Imkeller and Müller (2005), Heunis (2015) and Cheridito and Hu (2011) for closed constraints. On the other hand, for recursive utilities, in a market with stochastic investment opportunities, the analysis was developed by El Karoui, Peng and Quenez (2001), Wang, Wang and Yang (2016), Aurand and Huang (2020), Schroder and Skiadas (2003), Schroder and Skiadas (2005), Matoussi, Mezghani and Mnif (2015), Yang, Liang and Zhou (2019) and Melnyk, Muhle-Karbe and Seifried (2020).…”

Equilibrium strategies for time-inconsistent stochastic switching systems

“…See [41] and references cited therein for a detailed presentation. The readers are also referred to [37,5,6,47,16] for some results relevant to financial applications with regime-switching systems.…”