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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.

MSC:

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
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