On the role of F\"ollmer-Schweizer minimal martingale measure in Risk Sensitive control Asset Management

TL;DR

This paper investigates the significance of the Föllmer-Schweizer minimal martingale measure in risk-sensitive asset management, revealing conditions under which the optimal strategy simplifies to investing solely in the riskless asset.

Contribution

It explicitly links the optimal wealth allocation to the minimal martingale measure using a stochastic game framework, providing new insights into portfolio strategies under model uncertainty.

Findings

Optimal strategy is to invest only in the riskless asset without short-selling under certain conditions.

When factor and stock processes have independent noise, the minimal martingale measure governs the factor process.

The paper clarifies the role of the minimal martingale measure in portfolio optimization under risk sensitivity.

Abstract

Kuroda and Nagai \cite{KN} state that the factor process in the Risk Sensitive control Asset Management (RSCAM) is stable under the F\"ollmer-Schweizer minimal martingale measure . Fleming and Sheu \cite{FS} and more recently F\"ollmer and Schweizer \cite{FoS} have observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this article we aim to address this question by explicitly connecting the optimal wealth allocation to the minimal martingale measure. We achieve this by using a "trick" of observing this problem in the context of model uncertainty via a two person zero sum stochastic differential game between the investor and an antagonistic market that provides a probability measure. We obtain some startling insights. Firstly, if short-selling is not permitted and if the factor process evolves under the minimal martingale measure then the investor's optimal strategy can only be to invest in the riskless asset (i.e. the no-regret strategy). Secondly, if the factor process and the stock price process have independent noise, then even if the market allows short selling, the optimal strategy for the investor must be the no-regret strategy while the factor process will evolve under the minimal martingale measure .