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On Optimal Terminal Wealth Problems with Random Trading Times and Drawdown Constraints

Part of: Stochastic processes Stochastic systems and control

Published online by Cambridge University Press:  22 February 2016

Ulrich Rieder  and
Marc Wittlinger
Ulrich Rieder*
Affiliation:
Ulm University
Marc Wittlinger*
Affiliation:
Ulm University
*
∗ Postal address: Institut für Optimierung und Operations Research, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany. Email address: [email protected]
∗∗ Postal address: Institute of Mathematical Finance, Ulm University, Helmholtzstrasse 18, 89081 Ulm, Germany. Email address: [email protected]
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Abstract

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We consider an investment problem where observing and trading are only possible at random times. In addition, we introduce drawdown constraints which require that the investor's wealth does not fall under a prior fixed percentage of its running maximum. The financial market consists of a riskless bond and a stock which is driven by a Lévy process. Moreover, a general utility function is assumed. In this setting we solve the investment problem using a related limsup Markov decision process. We show that the value function can be characterized as the unique fixed point of the Bellman equation and verify the existence of an optimal stationary policy. Under some mild assumptions the value function can be approximated by the value function of a contracting Markov decision process. We are able to use Howard's policy improvement algorithm for computing the value function as well as an optimal policy. These results are illustrated in a numerical example.

Keywords

Portfolio optimizationilliquid marketrandom trading timedrawdown constraintlimsup Markov decision processHoward's policy improvement algorithmLévy process

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60G51: Processes with independent increments; L'evy processes 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 121 - 138
Copyright
© Applied Probability Trust 

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