Optimal investment and consumption in a black-scholes market with lévy-driven stochastic coefficients

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.