Risk Sensitive Investment Management with Affine Processes: a Viscosity Approach

TL;DR

This paper develops a mathematical framework for risk-sensitive investment management using affine jump-diffusion models, proving the value function is the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation.

Contribution

It extends existing jump-diffusion models to include valuation factor jumps and establishes the viscosity solution property for the control problem's value function.

Findings

Proves the value function is the unique viscosity solution.

Extends models to include jumps in asset prices and valuation factors.

Provides a rigorous mathematical foundation for risk-sensitive optimization.

Abstract

In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. 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.) In this setting, the Hamilton- Jacobi-Bellman equation is a partial integro-differential PDE. The main result of the paper is to show that the value function of the control problem is the unique viscosity solution of the Hamilton-Jacobi-Bellman equation.