Jump-Diffusion Risk-Sensitive Asset Management

TL;DR

This paper addresses a risk-sensitive portfolio optimization problem with asset prices modeled by SDEs driven by Brownian motion and Poisson jumps, reducing it to a jump-free control problem and proving the existence of a classical solution to the associated HJB equation.

Contribution

It introduces a change of measure technique to simplify the jump-diffusion control problem and proves the HJB equation has a classical solution, advancing theoretical understanding in risk-sensitive asset management.

Findings

Reduction of jump-diffusion problem to a jump-free control problem

Existence of a classical solution to the HJB equation

Application of policy improvement and PDE techniques

Abstract

This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion 'factor' process. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance.) By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of the paper is that the Hamilton-Jacobi-Bellman equation for this problem has a classical solution. The proof uses Bellman's "policy improvement" method together with results on linear parabolic PDEs due to Ladyzhenskaya et al.