The Annals of Applied Probability

Optimal stopping problems for the maximum process with upper and lower caps

Curdin Ott

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This paper concerns optimal stopping problems driven by the running maximum of a spectrally negative Lévy process $X$. More precisely, we are interested in modifications of the Shepp–Shiryaev optimal stopping problem [Avram, Kyprianou and Pistorius Ann. Appl. Probab. 14 (2004) 215–238; Shepp and Shiryaev Ann. Appl. Probab. 3 (1993) 631–640; Shepp and Shiryaev Theory Probab. Appl. 39 (1993) 103–119]. First, we consider a capped version of the Shepp–Shiryaev optimal stopping problem and provide the solution explicitly in terms of scale functions. In particular, the optimal stopping boundary is characterised by an ordinary differential equation involving scale functions and changes according to the path variation of $X$. Secondly, in the spirit of [Shepp, Shiryaev and Sulem Advances in Finance and Stochastics (2002) 271–284 Springer], we consider a modification of the capped version of the Shepp–Shiryaev optimal stopping problem in the sense that the decision to stop has to be made before the process $X$ falls below a given level.

Article information

Ann. Appl. Probab., Volume 23, Number 6 (2013), 2327-2356.

First available in Project Euclid: 22 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G51: Processes with independent increments; Lévy processes 60J75: Jump processes

Optimal stopping optimal stopping boundary principle of smooth fit principle of continuous fit Lévy processes scale functions


Ott, Curdin. Optimal stopping problems for the maximum process with upper and lower caps. Ann. Appl. Probab. 23 (2013), no. 6, 2327--2356. doi:10.1214/12-AAP903.

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  • [1] Alili, L. and Kyprianou, A. E. (2005). Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15 2062–2080.
  • [2] Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Probab. 14 215–238.
  • [3] Baurdoux, E. J. and Kyprianou, A. E. (2009). The Shepp–Shiryaev stochastic game driven by a spectrally negative Lévy process. Theory Probab. Appl. 53 481–499.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [5] Bichteler, K. (2002). Stochastic Integration with Jumps. Encyclopedia of Mathematics and Its Applications 89. Cambridge Univ. Press, Cambridge.
  • [6] Carr, P. and Wu, L. (2003). The finite moment log stable process and option pricing. J. Finance 58 753–778.
  • [7] Chan, T. (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab. 9 504–528.
  • [8] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th ed. Elsevier, Amsterdam.
  • [9] Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2011). The theory of scale functions for spectrally negative Lévy processes. Available at arXiv:1104.1280v1 [math.PR].
  • [10] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • [11] Madan, D. B. and Schoutens, W. (2008). Break on through to the single side. J. Credit Risk 4(3).
  • [12] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econ. Stat. 1 247–257.
  • [13] Mikalevich, V. S. (1958). Baysian choice between two hypotheses for the mean value of a normal process. Visn. Kiiv. Univ. Ser. Fiz.-Mat. Nauki 1 101–104.
  • [14] Peskir, G. (1998). Optimal stopping of the maximum process: The maximality principle. Ann. Probab. 26 1614–1640.
  • [15] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • [16] Peskir, G. and Shiryaev, A. N. (2000). Sequential testing problems for Poisson processes. Ann. Statist. 28 837–859.
  • [17] Pistorius, M. R. (2004). On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theoret. Probab. 17 183–220.
  • [18] Podlubny, I. (2012). Mittag-Leffler function [Matlab code]. Available at
  • [19] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
  • [20] Shepp, L. and Shiryaev, A. N. (1993). The Russian option: Reduced regret. Ann. Appl. Probab. 3 631–640.
  • [21] Shepp, L. A. and Shiryaev, A. N. (1993). A new look at pricing of the “Russian option.” Theory Probab. Appl. 39 103–119.
  • [22] Shepp, L. A., Shiryaev, A. N. and Sulem, A. (2002). A barrier version of the Russian option. In Advances in Finance and Stochastics 271–284. Springer, Berlin.
  • [23] Shiryaev, A. N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. Advanced Series on Statistical Science & Applied Probability 3. World Scientific, River Edge, NJ.