## Journal of Applied Probability

- J. Appl. Probab.
- Volume 52, Number 3 (2015), 703-717.

### On the role of Föllmer-Schweizer minimal martingale measure in risk-sensitive control asset management

#### Abstract

Kuroda and Nagai (2002) stated that the factor process in risk-sensitive control asset management is stable under the Föllmer-Schweizer minimal martingale measure. Fleming and Sheu (2002) and, more recently, Föllmer and Schweizer (2010) observed that the role of the minimal martingale measure in this portfolio optimization is yet to be established. In this paper 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 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.

#### Article information

**Source**

J. Appl. Probab., Volume 52, Number 3 (2015), 703-717.

**Dates**

First available in Project Euclid: 22 October 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1445543841

**Digital Object Identifier**

doi:10.1239/jap/1445543841

**Mathematical Reviews number (MathSciNet)**

MR3414986

**Zentralblatt MATH identifier**

1346.60102

**Subjects**

Primary: 49L02

Secondary: 60G02

**Keywords**

Risk-sensitive control asset management minimal martingale measure zero sum stochastic differential game stability

#### Citation

Deshpande, Amogh. On the role of Föllmer-Schweizer minimal martingale measure in risk-sensitive control asset management. J. Appl. Probab. 52 (2015), no. 3, 703--717. doi:10.1239/jap/1445543841. https://projecteuclid.org/euclid.jap/1445543841