Annals of Applied Probability

Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients

Łukasz Delong and Claudia Klüppelberg

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Abstract

In this paper, we investigate an optimal investment and consumption problem for an investor who trades in a Black–Scholes financial market with stochastic coefficients driven by a non-Gaussian Ornstein–Uhlenbeck process. We assume that an agent makes investment and consumption decisions based on a power utility function. By applying the usual separation method in the variables, we are faced with the problem of solving a nonlinear (semilinear) first-order partial integro-differential equation. A candidate solution is derived via the Feynman–Kac representation. By using the properties of an operator defined in a suitable function space, we prove uniqueness and smoothness of the solution. Optimality is verified by applying a classical verification theorem.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 3 (2008), 879-908.

Dates
First available in Project Euclid: 26 May 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1211819788

Digital Object Identifier
doi:10.1214/07-AAP475

Mathematical Reviews number (MathSciNet)
MR2418232

Zentralblatt MATH identifier
1140.93048

Subjects
Primary: 93E20: Optimal stochastic control 91B28
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60J75: Jump processes

Keywords
Banach fixed point theorem Feynman–Kac formula Hamilton–Jacobi–Bellman equation utility function Lévy process optimal investment and consumption Ornstein–Uhlenbeck process stochastic volatility model subordinator

Citation

Delong, Łukasz; Klüppelberg, Claudia. Optimal investment and consumption in a Black–Scholes market with Lévy-driven stochastic coefficients. Ann. Appl. Probab. 18 (2008), no. 3, 879--908. doi:10.1214/07-AAP475. https://projecteuclid.org/euclid.aoap/1211819788


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