# Predicting the Time at Which a Lévy Process Attains Its Ultimate Supremum

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## Abstract

We consider the problem of finding a stopping time that minimises the *L* ^{1}-distance to *θ*, the time at which a Lévy process attains its ultimate supremum. This problem was studied in Du Toit and Peskir (Proc. Math. Control Theory Finance, pp. 95–112, 2008) for a Brownian motion with drift and a finite time horizon. We consider a general Lévy process and an infinite time horizon (only compound Poisson processes are excluded. Furthermore due to the infinite horizon the problem is interesting only when the Lévy process drifts to −∞). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If *θ* has infinite mean there exists no stopping time with a finite *L* ^{1}-distance to *θ*, whereas if *θ* has finite mean it is either optimal to stop immediately or to stop when the process reflected in its supremum exceeds a positive level, depending on whether the median of the law of the ultimate supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the Lévy process has no positive jumps.

## Keywords

Lévy processes Optimal prediction Optimal stopping## Mathematics Subject Classification

60G40 62M20## Notes

### Acknowledgements

The authors are very grateful to Jenny Sexton for useful suggestions and discussion as well as to two anonymous referees for the helpful comments.

## References

- 1.Alili, L., Kyprianou, A.E.: Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab.
**15**, 2062–2080 (2005) CrossRefzbMATHMathSciNetGoogle Scholar - 2.Allaart, P.: A general ‘bang-bang’ principle for predicting the maximum of a random walk. J. Appl. Probab.
**47**, 1072–1083 (2010) CrossRefzbMATHMathSciNetGoogle Scholar - 3.Baurdoux, E.J., Van Schaik, K.: Further calculations for the McKean stochastic game for a spectrally negative Lévy process: from a point to an interval. J. Appl. Probab.
**48**, 200–216 (2011) CrossRefzbMATHMathSciNetGoogle Scholar - 4.Bernyk, V., Dalang, R.C., Peskir, G.: Predicting the ultimate supremum of a stable Lévy process with no negative jumps. Ann. Probab.
**39**, 2385–2423 (2011) CrossRefzbMATHMathSciNetGoogle Scholar - 5.Bertoin, J.: Sur la décomposition de la trajectoire d’un processus de Lévy spectralement positif en son infimum. Ann. Inst. Henri Poincaré, B Calc. Probab. Stat.
**4**, 537–547 (1991) MathSciNetGoogle Scholar - 6.Bertoin, J.: Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stoch. Process. Appl.
**47**, 17–35 (1993) CrossRefzbMATHMathSciNetGoogle Scholar - 7.Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
- 8.Bichteler, K.: Stochastic Integration with Jumps. Cambridge University Press, Cambridge (2002) CrossRefzbMATHGoogle Scholar
- 9.Chaumont, L., Doney, R.A.: On Lévy processes conditioned to stay positive. Electron. J. Probab.
**10**, 948–961 (2005) CrossRefMathSciNetGoogle Scholar - 10.Chaumont, L.: On the law of the supremum of Lévy processes. Ann. Probab.
**41**, 1191–1217 (2013) CrossRefzbMATHMathSciNetGoogle Scholar - 11.Cohen, A.: Examples of optimal prediction in the infinite horizon case. Stat. Probab. Lett.
**80**, 950–957 (2010) CrossRefzbMATHGoogle Scholar - 12.Du Toit, J., Peskir, G.: The trap of complacency in predicting the maximum. Ann. Probab.
**35**, 340–365 (2007) CrossRefzbMATHMathSciNetGoogle Scholar - 13.Du Toit, J., Peskir, G.: Predicting the time of the ultimate maximum for Brownian motion with drift. In: Proc. Math. Control Theory Finance, pp. 95–112 (2008) CrossRefGoogle Scholar
- 14.Du Toit, J., Peskir, G.: Selling a stock at the ultimate maximum. Ann. Appl. Probab.
**19**, 983–1014 (2009) CrossRefzbMATHMathSciNetGoogle Scholar - 15.Duquesne, T.: Path decompositions for real Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat.
**39**(2), 339–370 (2003) CrossRefzbMATHMathSciNetGoogle Scholar - 16.Espinosa, G.-E., Touzi, N.: Detecting the maximum of a mean-reverting scalar diffusion. SIAM J. Control Optim.
**50**, 2543–2572 (2012) CrossRefzbMATHMathSciNetGoogle Scholar - 17.Glover, K., Hulley, H., Peskir, G.: Three-dimensional Brownian motion and the golden ratio rule. Ann. Appl. Probab.
**23**, 895–922 (2013) CrossRefzbMATHMathSciNetGoogle Scholar - 18.Graversen, S.E., Peskir, G., Shiryaev, A.N.: Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Probab. Appl.
**45**, 125–136 (2001) CrossRefMathSciNetGoogle Scholar - 19.Hubalek, F., Kyprianou, A.E.: Old and new examples of scale functions for spectrally negative Lévy processes. In: Dalang, R., Dozzi, M., Russo, F. (eds.) Sixth Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, pp. 119–146. Birkhäuser, Basel (2010) Google Scholar
- 20.Kleinert, F., Van Schaik, K.: A variation of the Canadisation algorithm for the pricing of American options driven by Lévy processes. Submitted (2013). arXiv:1304.4534
- 21.Kuznetsov, A., Kyprianou, A.E., Pardo, J.C.: Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab.
**22**, 1101–1135 (2012) CrossRefzbMATHMathSciNetGoogle Scholar - 22.Kyprianou, A.E.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006) zbMATHGoogle Scholar
- 23.Lambert, A.: Completely asymmetric Lévy processes confined in a finite interval. Ann. Inst. Henri Poincaré Probab. Stat.
**36**, 251–274 (2000) CrossRefzbMATHGoogle Scholar - 24.Levendorskii, S.Z., Kudryavtsev, O., Zherder, V.: The relative efficiency of numerical methods for pricing American options under Lévy processes. J. Comput. Finance
**9**, 69–97 (2006) Google Scholar - 25.Maller, R.A., Solomon, D.H., Szimayer, A.: A multinomial approximation for American option prices in Lévy process models. Math. Finance
**16**, 613–633 (2006) CrossRefzbMATHMathSciNetGoogle Scholar - 26.Matache, A.M., Nitsche, P.A., Schwab, C.: Wavelet Galerkin pricing of American options on Lévy driven assets. Quant. Finance
**5**, 403–424 (2005) CrossRefzbMATHMathSciNetGoogle Scholar - 27.Millar, P.W.: Zero-one laws and the minimum of a Markov process. Trans. Am. Math. Soc.
**226**, 365–391 (1977) CrossRefzbMATHMathSciNetGoogle Scholar - 28.Mordecki, E.: The distribution of the maximum of a Lévy process with positive jumps of phase-type. In: Proceedings of the Conference Dedicated to the 90th Anniversary of Boris Vladimirovich Gnedenko, Kyiv, pp. 309–316 (2002) Google Scholar
- 29.Peskir, G.: Optimal detection of a hidden target: the median rule. Stoch. Process. Appl.
**122**, 2249–2263 (2012) CrossRefzbMATHMathSciNetGoogle Scholar - 30.Peskir, G., Shiryaev, A.N.: Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel (2006) zbMATHGoogle Scholar
- 31.Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
- 32.Urusov, M.A.: On a property of the moment at which Brownian motion attains its maximum and some optimal stopping problems. Theory Probab. Appl.
**49**, 169–176 (2005) CrossRefMathSciNetGoogle Scholar - 33.Vigon, V.: Votre Lévy rampe-t-il? J. Lond. Math. Soc.
**65**, 243–256 (2002) CrossRefzbMATHMathSciNetGoogle Scholar