STABILITY OF MERTON'S PORTFOLIO OPTIMIZATION PROBLEM FOR LÉVY MODELS

Abstract

Merton's classical portfolio optimisation problem for an investor, who can trade in a risk-free bond and a stock, can be extended to the case where the driving noise of the log-returns is a pure jump process instead of a Brownian motion. Benth et al. [5], [6] solved the problem and found in the HARA-utility case the optimal control implicitly given by an integral equation. There are several ways to approximate a Levy process with infinite activity: by neglecting the small jumps or approximating them with a Brownian motion, as discussed in Asmussen and Rosinski [2]. In this setting, we study stability of the corresponding optimal investment problems. The optimal controls are solutions of integral equations, for which we study convergence. We are able to characterize the rate of convergence in terms of the variance of the small jumps. Additionally, we prove convergence of the corresponding wealth processes and indirect utilities (value functions).