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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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

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

First available in Project Euclid: 26 May 2008

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Zentralblatt MATH identifier

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

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


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.

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  • [1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Univ. Press.
  • [2] Bensoussan, A. and Lions, J. L. (1984). Impulse Control and Quasi-Variational Inequalities. Bordas, Paris.
  • [3] Bäuerle, N. and Rieder, U. (2005). Portfolio optimization with unobservable Markov modulated drift process. J. Appl. Probab. 42 362–378.
  • [4] Becherer, D. and Schweizer, M. (2005). Classical solutions to reaction–diffusion systems for hedging problems with interacting Itô and point processes. Ann. Appl. Probab. 15 1111–1155.
  • [5] Benth, F. E., Karlsen, K. H. and Reikvam, K. (2003) Merton’s portfolio optimization problem in Black–Scholes market with non-Gaussian stochastic volatility of Ornstein–Uhlenbeck type. Math. Finance 13 215–244.
  • [6] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial mathematics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
  • [7] Bertoin, J. (1996). Lévy Processes. Cambridge Univ. Press.
  • [8] Castañeda-Leyva, N. and Hernández-Hernández, D. (2005). Optimal consumption investment problems in incomplete markets with stochastic coefficient. SIAM J. Control Optim. 44 1322–1344.
  • [9] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman and Hall/CRC, Boca Raton, FL.
  • [10] Delong, Ł. (2006). Optimal investment and consumption in the presence of default in a financial market driven by a Lévy process. Ann. Univ. Mariae Curie-Sklodowska Sect. A 60 1–15.
  • [11] Delong, Ł. (2007). Optimal investment strategies in financial markets driven by a Lévy process, with applications to insurance. Ph.D. thesis, at The Institute of Mathematics of the Polish Academy of Sciences.
  • [12] Fleming, W. H. and Hernández-Hernández, D. (2003). An optimal consumption model with stochastic volatility. Finance Stoch. 7 245–262.
  • [13] Fleming, W. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, Berlin.
  • [14] Goll, T. and Kallsen, J. (2000). Optimal portfolios for logarithmic utility. Stochastic Process. Appl. 89 31–48.
  • [15] Hernández-Hernández, D. and Schied, A. (2006). Robust utility maximization in a stochastic factor model. Statist. Decisions 24 109–125.
  • [16] Kraft, H. and Steffensen, M. (2006). Portfolio problems stopping at first hitting time with applications to default risk. Math. Methods Oper. Res. 63 123–150.
  • [17] Lindberg, C. (2006). News-generated dependency and optimal portfolios for n stocks in a market of Barndorff-Nielsen and Shephard type. Math. Finance 16 549–568.
  • [18] Merton, R. C. (1971). Optimal consumption and portfolio rules in continuous time model. J. Econom. Theory 3 373–413.
  • [19] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump-Diffusions, 2nd ed. Springer, Berlin.
  • [20] Pham, H. (1998). Optimal stopping of controlled jump-diffusions: A viscosity solution approach. J. Math. Sys. Estim. Control 8 1–27.
  • [21] Pham, H. and Quenez, M. C. (2001). Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11 210–238.
  • [22] Sato, K.-I. (1999). Lévy Processes and Infinite Divisibility. Cambridge Univ. Press.
  • [23] Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risk. Finance Stoch. 5 61–82.