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

Published online by Cambridge University Press:  30 March 2016

Amogh Deshpande
Amogh Deshpande*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK. Email address: amogh.deshpande@warwick.ac.uk, addeshpa@gmail.com
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Abstract

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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.

Keywords

Risk-sensitive control asset managementminimal martingale measurezero sum stochastic differential gamestability
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 3 , September 2015 , pp. 703 - 717
Copyright
Copyright © 2015 by the Applied Probability Trust 

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