An Optimal Investment Problem under Correlated Noises: Risk-Sensitive Stochastic Control Approach

TL;DR

This paper develops a risk-sensitive stochastic control framework to solve an optimal investment problem with correlated noises, deriving explicit strategies and analyzing their sensitivity to key parameters.

Contribution

It introduces a new stochastic maximum principle for correlated noises and provides explicit solutions for the optimal investment strategy under risk-seeking conditions.

Findings

Explicit optimal investment strategy in feedback form

Sensitivity analysis of strategies to risk parameter and correlation

Numerical simulations illustrating strategy behavior

Abstract

This paper is concerned with an optimal investment problem under correlated noises in the financial market, and the expected utility functional is hyperbolic absolute risk aversion (HARA) with the exponent $\gamma\neq0$. The problem can be reformulated as a risk-sensitive stochastic control problem. A new stochastic maximum principle is obtained first, where the adjoint equations and maximum condition heavily depend on the risk-sensitive parameter and the correlation coefficient. The optimal investment strategy is obtained explicitly in a state feedback form via the solution to a certain Riccati equation, under the risk-seeking case. Numerical simulation and figures are given to illustrate the sensitivity for the optimal investment strategy, with respect to the risk-sensitive parameter and the correlation coefficient.