Utility maximization in constrained and unbounded financial markets: Applications to indifference valuation, regime switching, consumption and Epstein-Zin recursive utility

Abstract: This memoir presents a systematic study of the utility maximization problem of an investor in a constrained and unbounded financial market. Building upon the work of Hu et al. (2005) [Ann. Appl. Probab., 15, 1691--1712] in a bounded framework, we extend our analysis to the more challenging unbounded case. Our methodology combines both methods of quadratic backward stochastic differential equations with unbounded solutions and convex duality. Central to our approach is the verification of the finite entropy condition, which plays a pivotal role in solving the underlying utility maximization problem and establishing the martingale property and the convex duality representation of the conditional value process. Through four distinct applications, we first study the utility indifference valuation of financial derivatives with unbounded payoffs, uncovering novel asymptotic behaviors as the risk aversion parameter approaches zero or infinity. Furthermore, we study the regime switching market model with unbounded random endowments and consumption-investment problems with unbounded random endowments, both constrained to portfolios chosen from a convex and closed set. Finally, we investigate the investment-consumption problem of an investor with the Epstein-Zin recursive utility in an unbounded financial market.