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

Part of: Special processes Mathematical finance Stochastic systems and control 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: ekstrom@math.uu.se
*Postal address: Uppsala University, Box 480, 75106 Uppsala, Sweden. Email address: ekstrom@math.uu.se
**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

Information

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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References

Akian, M., Sulem, A. and Taksar, M. (2001). Dynamic optimization of long-term growth rate for a portfolio with transaction costs and logarithmic utility. Math. Finance 11, 153–188.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Applications of Mathematics (New York), Vol. 51, Stochastic Modelling and Applied Probability. Springer, New York.Google Scholar
Bayraktar, E. and Egami, M. (2010). On the one-dimensional optimal switching problem. Math. Operat. Res. 35, 140–159.10.1287/moor.1090.0432CrossRefGoogle Scholar
Beneš, V. E. and Karatzas, I. (1981). Optimal stationary linear control of the Wiener process. J. Optimization Theory Appl. 35, 611–633.10.1007/BF00934934CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion–Facts and Formulae, 2nd edn. Birkhäuser, Basel.10.1007/978-3-0348-8163-0CrossRefGoogle Scholar
Brekke, K. A. and Øksendal, B. (1994). Optimal switching in an economic activity under uncertainty. SIAM J. Control Optimization 32, 1021–1036.10.1137/S0363012992229835CrossRefGoogle Scholar
Browne, S. and Whitt, W. (1996). Portfolio choice and the Bayesian Kelly criterion. Adv. Appl. Prob. 28, 1145–1176.10.2307/1428168CrossRefGoogle Scholar
Dai, M., Zhang, Q. and Zhu, Q. (2010). Trend-following trading under a regime switching model. SIAM J. Financial Math. 1, 780–810.10.1137/090770552CrossRefGoogle Scholar
Dai, M., Yang, Z., Zhang, Q. and Zhu, Q. (2016). Optimal trend-following trading rules. Math. Operat. Res. 41, 626–642.10.1287/moor.2015.0743CrossRefGoogle Scholar
Duckworth, K. and Zervos, M. (2001). A model for investment decisions with switching costs. Ann. Appl. Prob. 11, 239–260.10.1214/aoap/998926992CrossRefGoogle Scholar
Gut, A. and Janson, S. (1983). The limiting behaviour of certain stopped sums and some applications. Scand. J. Statist. 10, 281–292.Google Scholar
Karatzas, I. (1983). A class of singular stochastic control problems. Adv. Appl. Prob. 15, 225–254.10.1017/S0001867800021169CrossRefGoogle Scholar
Kelly, J. L. Jr. (1956). A new interpretation of information rate. Bell. Syst. Tech. J. 35, 917–926.10.1002/j.1538-7305.1956.tb03809.xCrossRefGoogle Scholar
Liptser, R. and Shiryaev, A. (2001). Statistics of Random Processes. I. General Theory, 2nd edn. Applications of Mathematics, Vol. 5. Stochastic Modelling and Applied Probability. Springer, Berlin.Google Scholar
Ross, S. (1996). Stochastic Processes, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley, New York.Google Scholar
Zeng, Z. and Lee, C.-G. (2014). Pairs trading: optimal thresholds and profitability. Quant. Finance 14, 1881–1893.10.1080/14697688.2014.917806CrossRefGoogle Scholar