Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model
Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model
This paper addresses a finite-horizon risk-sensitive portfolio optimization problem where asset prices follow jump-diffusion processes with drifts depending on an auxiliary diffusion factor process. The investor aims to maximize the risk-adjusted growth rate, parameterized by a risk sensitivity parameter. The key innovation is the extension of Kuroda and Nagai's change of measure technique to jump-diffusion settings, which transforms the original problem into a stochastic control problem for a pure diffusion process. This transformation eliminates the jump terms from the dynamics under the new measure, leading to a Hamilton-Jacobi-Bellman (HJB) equation that is a parabolic partial differential equation (PDE) rather than a partial integro-differential equation (PIDE). The paper proves that this HJB PDE admits a unique classical C^{1,2} solution, establishing existence via a policy improvement algorithm and PDE theory, and uniqueness via a verification argument. A critical technical contribution is the proof that the optimal control is admissible, requiring the Doléans exponential to be a martingale, which is nontrivial in the presence of jumps. The results hold under natural assumptions, including infinite activity jumps and correlation between factor and asset risks, without requiring strong boundedness conditions. This work significantly generalizes earlier diffusion-only models and provides a rigorous foundation for risk-sensitive asset management with jump risks.
Highlights
- 1Extends risk-sensitive asset management to jump-diffusion asset prices with correlated factor and asset risks.
- 2Shows that the jump-diffusion problem reduces to a diffusion control problem via a change of measure.
- 3Characterizes the value function by a parabolic HJB PDE rather than a PIDE, admitting a classical C^{1,2} solution.
- 4Proves admissibility of the optimal control by establishing martingale property of the Doléans exponential.
- 5Provides existence and uniqueness of the classical solution via policy improvement and PDE theory.
Methods
- MRisk-sensitive control with exponential-of-integral criterion.
- MChange of measure technique (Kuroda and Nagai) to transform to a diffusion problem.
- MDynamic programming and Hamilton-Jacobi-Bellman (HJB) PDE analysis.
- MPolicy improvement algorithm and linear parabolic PDE theory for existence of classical solutions.
Results
- RThe optimal control problem reduces to a diffusion control problem with no jumps.
- RThe HJB equation is a parabolic PDE (not PIDE) and admits a unique classical C^{1,2} solution.
- RThe optimal control is admissible, i.e., the Doléans exponential is a martingale.
- RExistence and uniqueness of the solution hold under natural assumptions without strong boundedness conditions.
- RThe framework allows for infinite activity jumps and correlation between factor and asset risks.
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