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A renewal theory approach to two-state switching problems with infinite values

Part of: Mathematical finance Stochastic systems and control Special processes Sequential methods

Published online by Cambridge University Press:  04 May 2020

Erik Ekström ,
Marcus Olofsson  and
Martin Vannestål
Erik Ekström*
Affiliation:
Uppsala University
Marcus Olofsson*
Affiliation:
Uppsala University
Martin Vannestål*
Affiliation:
Uppsala University
*
*Postal address: Uppsala University, Box 480, 75106 Uppsala, Sweden. Email address: [email protected]
*Postal address: Uppsala University, Box 480, 75106 Uppsala, Sweden. Email address: [email protected]
**Since the completion of a first draft of the current paper, our coauthor Martin Vannestål tragically passed away. In sorrow, we dedicate this work to his memory.
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Abstract

We study a renewal theory approach to perpetual two-state switching problems with infinite value functions. Since the corresponding value functions are infinite, the problems fall outside the standard class of problems which can be analyzed using dynamic programming. Instead, we propose an alternative formulation of optimal switching theory in which optimality of a strategy is defined in terms of its long-term mean return, which can be determined using renewal theory. The approach is illustrated by examples in connection with trend-following strategies in finance.

Keywords

Optimal switchingrenewal theoryexpected growth ratetrend following

MSC classification

Primary: 60K05: Renewal theory 93E20: Optimal stochastic control
Secondary: 62L15: Optimal stopping 91G10: Portfolio theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 1 - 18
Copyright
© Applied Probability Trust 2020

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