Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model

TL;DR

This paper develops a method to solve risk-sensitive asset management problems with jump-diffusion models by reducing them to classical PDEs, enabling more tractable analysis and solutions.

Contribution

It introduces a novel approach to characterize risk-sensitive jump-diffusion problems using classical PDEs instead of PIDEs, simplifying the analysis.

Findings

The risk-sensitive problem is fully characterized by a classical HJB PDE.

The PDE admits a classical C^{1,2} solution.

The approach simplifies the analysis of jump-diffusion risk-sensitive control problems.

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 to show that the risk-sensitive jump diffusion problem can be fully characterized in terms of a parabolic Hamilton-Jacobi-Bellman PDE rather than a PIDE, and that this PDE admits a classical C^{1,2} solution.