×

Optimal stopping problems for some Markov processes. (English) Zbl 1246.60065

Summary: We solve explicitly the optimal stopping problem with random discounting and an additive functional as cost of observations for a regular linear diffusion. We also extend the results to the class of one-sided regular Feller processes. This generalizes the result of M. Beibel and H. R. Lerche [Stat. Sin. 7, No. 1, 93–108 (1997; Zbl 0895.60048); Theory Probab. Appl. 45, No. 4, 547–557 (2000) and Teor. Veroyatn. Primen. 45, No. 4, 657–669 (2000; Zbl 0994.60046)] and A. Irle and V. Paulsen [Sequential Anal. 23, No. 3, 297–316 (2004; Zbl 1146.60307)]. Our approach relies on a combination of techniques borrowed from potential theory and stochastic calculus. We illustrate our results by detailing some new examples ranging from linear diffusions to Markov processes of the spectrally negative type.

MSC:

60G40 Stopping times; optimal stopping problems; gambling theory
60J60 Diffusion processes
60J75 Jump processes (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[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. · Zbl 1083.60034 · doi:10.1214/105051605000000377
[2] Bally, V. and Stoica, L. (1987). A class of Markov processes which admit local times. Ann. Probab. 15 241-262. · Zbl 0615.60069 · doi:10.1214/aop/1176992266
[3] Baurdoux, E. J. (2007). Examples of optimal stopping via measure transformation for processes with one-sided jumps. Stochastics 79 303-307. · Zbl 1111.60023 · doi:10.1080/17442500600856297
[4] Beibel, M. and Lerche, H. R. (1997). A new look at optimal stopping problems related to mathematical finance. Statist. Sinica 7 93-108. · Zbl 0895.60048
[5] Beibel, M. and Lerche, H. R. (2000). A note on optimal stopping of regular diffusions under random discounting. Teor. Veroyatn. Primen. 45 657-669. · Zbl 0994.60046 · doi:10.1137/S0040585X9797852X
[6] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121 . Cambridge Univ. Press, Cambridge. · Zbl 0861.60003
[7] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Pure and Applied Mathematics 29 . Academic Press, New York. · Zbl 0169.49204
[8] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion-Facts and Formulae , 2nd ed. Birkhäuser, Basel. · Zbl 1012.60003
[9] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion 73-130. Rev. Mat. Iberoam., Madrid. · Zbl 0905.60056
[10] Dayanik, S. and Karatzas, I. (2003). On the optimal stopping problem for one-dimensional diffusions. Stochastic Process. Appl. 107 173-212. · Zbl 1075.60524 · doi:10.1016/S0304-4149(03)00076-0
[11] Dynkin, E. B. (1963). Optimal choice of the stopping moment of a Markov process. Dokl. Akad. Nauk SSSR 150 238-240. · Zbl 0242.60018
[12] Graversen, S. E. and Peškir, G. (1997). On Wald-type optimal stopping for Brownian motion. J. Appl. Probab. 34 66-73. · Zbl 0876.60023 · doi:10.2307/3215175
[13] Hawkes, J. (1979). Potential theory of Lévy processes. Proc. Lond. Math. Soc. (3) 38 335-352. · Zbl 0401.60069 · doi:10.1112/plms/s3-38.2.335
[14] Irle, A. and Paulsen, V. (2004). Solving problems of optimal stopping with linear costs of observations. Sequential Anal. 23 297-316. · Zbl 1146.60307 · doi:10.1081/SQA-200027048
[15] Itô, K. and McKean, H. P. (1996). Diffusion Processes and Their Sample Paths . Springer, Berlin. · Zbl 0837.60001
[16] Khoshnevisan, D., Salminen, P. and Yor, M. (2006). A note on a.s. finiteness of perpetual integral functionals of diffusions. Electron. Commun. Probab. 11 108-117 (electronic). · Zbl 1111.60061 · doi:10.1214/ECP.v11-1203
[17] Kramkov, D. O. and Mordetski, È. (1994). An integral option. Teor. Veroyatn. Primen. 39 201-211. · Zbl 0836.90012
[18] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications . Springer, Berlin. · Zbl 1104.60001
[19] Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Probab. 13 1077-1098. · Zbl 1039.60044 · doi:10.1214/aoap/1060202835
[20] Lamperti, J. (1967). On random time substitutions and the Feller property. In Markov Processes and Potential Theory ( Proc. Sympos. Math. Res. Center , Madison , Wis. , 1967) 87-101. Wiley, New York. · Zbl 0189.51601
[21] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205-225. · Zbl 0274.60052 · doi:10.1007/BF00536091
[22] Lebedev, N. N. (1972). Special Functions and Their Applications . Dover, New York. · Zbl 0271.33001
[23] Patie, P. (2008). q -invariant functions for some generalizations of the Ornstein-Uhlenbeck semigroup. ALEA Lat. Am. J. Probab. Math. Stat. 4 31-43. · Zbl 1168.60011
[24] Patie, P. (2009). Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration. Bull. Sci. Math. 133 355-382. · Zbl 1171.60009 · doi:10.1016/j.bulsci.2008.10.001
[25] Patie, P. (2009). Infinite divisibility of solutions to some self-similar integro-differential equations and exponential functionals of Lévy processes. Ann. Inst. Henri Poincaré Probab. Stat. 45 667-684. · Zbl 1180.31010 · doi:10.1214/08-AIHP182
[26] Patie, P. and Vigon, V. (2011). One-dimensional completely asymmetric Markov processes. Unpublished manuscript, ULB.
[27] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems . Birkhäuser, Basel. · Zbl 1115.60001
[28] Pitman, J. and Yor, M. (1981). Bessel processes and infinitely divisible laws. In Stochastic Integrals ( Proc. Sympos. , Univ. Durham , Durham , 1980). Lecture Notes in Math. 851 285-370. Springer, Berlin. · Zbl 0469.60076
[29] Prabhakar, T. R. (1971). A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math. J. 19 7-15. · Zbl 0221.45003
[30] Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 471-509. · Zbl 1077.60055 · doi:10.3150/bj/1120591185
[31] Rogers, L. C. G. and Williams, D. (2000). Diffusions , Markov Processes , and Martingales. Vol. 1. Cambridge Univ. Press, Cambridge. Reprint of the second (1994) edition. · Zbl 0826.60002
[32] Salminen, P. (1985). Optimal stopping of one-dimensional diffusions. Math. Nachr. 124 85-101. · Zbl 0594.60080 · doi:10.1002/mana.19851240107
[33] Shepp, L. A. (1967). A first passage problem for the Wiener process. Ann. Math. Statist. 38 1912-1914. · Zbl 0178.19402 · doi:10.1214/aoms/1177698626
[34] Shiryayev, A. N. (1978). Optimal Stopping Rules . Springer, New York. · Zbl 0391.60002
[35] Williams, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. Lond. Math. Soc. (3) 28 738-768. · Zbl 0326.60093 · doi:10.1112/plms/s3-28.4.738
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.