Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T00:03:52.699Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Bensoussan, A. (1992). Stochastic Control of Partially Observable Systems. Cambridge University Press.CrossRefGoogle Scholar
Berkovitz, L. D., Shreve, S. E. and Ziemer, W. P. (1993). A tribute to Wendell H. Fleming. J. SIAM Control Optimization 31 273-281.CrossRefGoogle Scholar
Bielecki, T. R. and Pliska, S. R. (1999). Risk-sensitive dynamic asset management. Appl. Math. Optimization 39 337-360.CrossRefGoogle Scholar
Fleming, W. H. (1995). Optimal investment models and risk sensitive stochastic control. In Mathematical Finance (IMA Vol. Math. Appl. 65), Springer, New York, pp. 75–88.CrossRefGoogle Scholar
Fleming, W. H. and Sheu, S. J. (2002). Risk-sensitive control and an optimal investment model. II. Ann. Appl. Prob. 12 730-767.Google Scholar
Föllmer, H. and Schweizer, M. (2010). Minimal martingale measure. In Encyclopedia of Quantitative Finance, John Wiley, New York, pp. 1200–1204.Google Scholar
Gīhman, I. Ī. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, New York.CrossRefGoogle Scholar
Girsanov, I. V. (1960). On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Prob. Appl. 5 285-301.Google Scholar
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.Google Scholar
Kuroda, K. and Nagai, H. (2002). Risk-sensitive portfolio optimization on infinite time horizon. Stoch. Stoch. Reports 73 309-331.CrossRefGoogle Scholar
Lefebvre, M. and Montulet, P. (1994). Risk-sensitive optimal investment policy. Internat. J. Systems Sci. 25 183-192.CrossRefGoogle Scholar
Whittle, P. (1990). Risk-Sensitive Optimal Control. John Wiley, Chichester.Google Scholar