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Optimality of doubly reflected Lévy processes in singular control

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  • Baurdoux, Erik J.
  • Yamazaki, Kazutoshi

Abstract

We consider a class of two-sided singular control problems. A controller either increases or decreases a given spectrally negative Lévy process so as to minimize the total costs comprising of the running and controlling costs where the latter is proportional to the size of control. We provide a sufficient condition for the optimality of a double barrier strategy, and in particular show that it holds when the running cost function is convex. Using the fluctuation theory of doubly reflected Lévy processes, we express concisely the optimal strategy as well as the value function using the scale function. Numerical examples are provided to confirm the analytical results.

Suggested Citation

  • Baurdoux, Erik J. & Yamazaki, Kazutoshi, 2015. "Optimality of doubly reflected Lévy processes in singular control," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2727-2751.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:7:p:2727-2751
    DOI: 10.1016/j.spa.2015.01.011
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    1. J. Michael Harrison & Michael I. Taksar, 1983. "Instantaneous Control of Brownian Motion," Mathematics of Operations Research, INFORMS, vol. 8(3), pages 439-453, August.
    2. Pistorius, M. R., 2003. "On doubly reflected completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 107(1), pages 131-143, September.
    3. Suresh P. Sethi & Michael I. Taksar, 2002. "Optimal Financing of a Corporation Subject To Random Returns," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 155-172, April.
    4. Egami, Masahiko & Leung, Tim & Yamazaki, Kazutoshi, 2013. "Default swap games driven by spectrally negative Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 347-384.
    5. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    6. George M. Constantinides & Scott F. Richard, 1978. "Existence of Optimal Simple Policies for Discounted-Cost Inventory and Cash Management in Continuous Time," Operations Research, INFORMS, vol. 26(4), pages 620-636, August.
    7. Florin Avram & Zbigniew Palmowski & Martijn R. Pistorius, 2007. "On the optimal dividend problem for a spectrally negative L\'{e}vy process," Papers math/0702893, arXiv.org.
    8. A. Kyprianou & B. Surya, 2007. "Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels," Finance and Stochastics, Springer, vol. 11(1), pages 131-152, January.
    9. Erhan Bayraktar & Masahiko Egami, 2008. "An Analysis of Monotone Follower Problems for Diffusion Processes," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 336-350, May.
    10. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2013. "On Optimal Dividends In The Dual Model," ASTIN Bulletin, Cambridge University Press, vol. 43(3), pages 359-372, September.
    11. D. Beyer & S. P. Sethi & M. Taksar, 1998. "Inventory Models with Markovian Demands and Cost Functions of Polynomial Growth," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 281-323, August.
    12. Agnès Sulem, 1986. "A Solvable One-Dimensional Model of a Diffusion Inventory System," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 125-133, February.
    13. Guo, Xin & Pham, Huyên, 2005. "Optimal partially reversible investment with entry decision and general production function," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 705-736, May.
    14. Eppen, Gary D & Fama, Eugene F, 1969. "Cash Balance and Simple Dynamic Portfolio Problems with Proportional Costs," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 10(2), pages 119-133, June.
    15. Hernández-Hernández, Daniel & Yamazaki, Kazutoshi, 2015. "Games of singular control and stopping driven by spectrally one-sided Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 1-38.
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    Cited by:

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