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In this paper, we investigate an optimal investment and consumption problemfor an investor who trades in a Black--Scholes financial market with stochasticcoefficients driven by a non-Gaussian Ornstein--Uhlenbeck process. We assumethat an agent makes investment and consumption decisions based on a powerutility function. By applying the usual separation method in the variables, weare faced with the problem of solving a nonlinear (semilinear) first-orderpartial integro-differential equation. A candidate solution is derived via theFeynman--Kac representation. By using the properties of an operator defined ina suitable function space, we prove uniqueness and smoothness of the solution.Optimality is verified by applying a classical verification theorem.

Optimal investment and consumption in a Black--Scholes market with L\'evy-driven stochastic coefficients