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

TL;DR

This paper develops a model for optimal investment and consumption in a Black--Scholes market with stochastic coefficients driven by a non-Gaussian Ornstein--Uhlenbeck process, deriving and verifying a unique solution to the associated complex equations.

Contribution

It introduces a novel approach to solving a nonlinear partial integro-differential equation for stochastic market coefficients using Feynman--Kac representation and operator theory.

Findings

Derived a candidate solution for the nonlinear equation

Proved uniqueness and smoothness of the solution

Verified optimality through a classical theorem

Abstract

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.