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A stochastic control approach to portfolio optimization with recursive utility in incomplete markets

Frank Seifried
(Technische Universität Kaiserslautern)
Thiele Seminar
Monday, 15 March, 2010, at 10:30-11:30, in Koll. D (1531-211)
We study the optimal portfolio and consumption decision of an investor with recursive preferences of Epstein-Zin-type in an incomplete market. Following a classical dynamic programming approach, we formulate the associated Hamilton-Jacobi-Bellman equation and provide a suitable verification theorem. The proof of this verification theoreom is complicated by the fact that the Epstein-Zin aggregator is non-Lipschitz, so standard verification results (e.g., in [Duffie, Epstein 1992]) are not applicable. We are then able to obtain explicit solutions for certain combinations of risk aversion and elasticity of intertemporal substitution (EIS) in the stochastic excess return model of [Wachter 2002] and the stochastic volatility models of [Liu 2007] and [Chacko, Viceira 2005]. Our results complement those of [Schroder, Skiadas 2003], who obtain explicit solutions for unit EIS with a utility gradient approach. Our contribution is twofold: First, we find explicit solutions for portfolio optimization problems with recursive preferences and non-unit EIS. Second, our approach is based on classical stochastic control methods and we provide a rigorous verification theorem.

[Chacko, Viceira 2005: Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets, Review of Financial Studies 18, 1369-1402]

[Duffie, Epstein 1992: Stochastic differential utility, Econometrica 60, 353-394]

[Liu 2007: Portfolio selection in stochastic environments, Review of Financial Studies 20, 1-39]

[Schroder, Skiadas 2003: Optimal lifetime portfolio-consumption strategies under trading constraints and generalized recursive preferences, Stochastic Processes and Their Applications 108, 155-202]

[Wachter 2002: Portfolio and consumption decisions under mean-reverting returns, Journal of Financial and Quantitative Analysis 37, 63-91]

Organised by: The T.N. Thiele Centre
Contact person: Søren Asmussen