Optimal investment and consumption in a Black-Scholes market with Lévy-driven stochastic coefficients. (English) Zbl 1140.93048

Summary: 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.


93E20 Optimal stochastic control
91G80 Financial applications of other theories
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J75 Jump processes (MSC2010)
47H10 Fixed-point theorems
49L20 Dynamic programming in optimal control and differential games
Full Text: DOI arXiv


[1] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus . Cambridge Univ. Press. · Zbl 1073.60002
[2] Bensoussan, A. and Lions, J. L. (1984). Impulse Control and Quasi-Variational Inequalities . Bordas, Paris. · Zbl 0324.49005
[3] Bäuerle, N. and Rieder, U. (2005). Portfolio optimization with unobservable Markov modulated drift process. J. Appl. Probab. 42 362-378. · Zbl 1138.93428
[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. · Zbl 1075.60080
[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. · Zbl 1049.91060
[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. JSTOR: · Zbl 0983.60028
[7] Bertoin, J. (1996). Lévy Processes . Cambridge Univ. Press. · Zbl 0861.60003
[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. · Zbl 1140.91381
[9] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes . Chapman and Hall/CRC, Boca Raton, FL. · Zbl 1052.91043
[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. · Zbl 1138.91431
[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. · Zbl 1035.60028
[13] Fleming, W. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control . Springer, Berlin. · Zbl 0323.49001
[14] Goll, T. and Kallsen, J. (2000). Optimal portfolios for logarithmic utility. Stochastic Process. Appl. 89 31-48. · Zbl 1048.91064
[15] Hernández-Hernández, D. and Schied, A. (2006). Robust utility maximization in a stochastic factor model. Statist. Decisions 24 109-125. · Zbl 1186.91229
[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. · Zbl 1136.91452
[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. · Zbl 1133.91431
[18] Merton, R. C. (1971). Optimal consumption and portfolio rules in continuous time model. J. Econom. Theory 3 373-413. · Zbl 1011.91502
[19] Øksendal, B. and Sulem, A. (2007). Applied Stochastic Control of Jump-Diffusions , 2nd ed. Springer, Berlin. · Zbl 1116.93004
[20] Pham, H. (1998). Optimal stopping of controlled jump-diffusions: A viscosity solution approach. J. Math. Sys. Estim. Control 8 1-27. · Zbl 0899.60039
[21] Pham, H. and Quenez, M. C. (2001). Optimal portfolio in partially observed stochastic volatility models. Ann. Appl. Probab. 11 210-238. · Zbl 1043.91032
[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. · Zbl 0977.93081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.