×

zbMATH — the first resource for mathematics

Optimal multiple stopping with random waiting times. (English) Zbl 1303.60032
This paper extends a theory of optimal multiple stopping. It is the optimization problem: \(\max_{(\tau_1\leq\dots\leq\tau_n)}{\mathbb E}(Y_{\tau_1}+\dots+Y_{\tau_n})\) (see [G. W. Haggstrom, Ann. Math. Stat. 38, 1618–1626 (1967; Zbl 0189.18301)] for the basic formulation) with multiple exercise opportunities. Without any further restrictions, the optimal strategy is to exercise all rights at that same time, namely, at the time when one would exercise for \(n = 1\). Typically, there are restrictions imposed on the exercise dates (see, e.g., [R. Carmona and N. Touzi, Math. Finance 18, No. 2, 239–268 (2008; Zbl 1133.91499)], i.e., restrictions that between each two exercise times there must be a given constant time interval \(\delta\)). In the article under review, random waiting times \(\delta_1,\dots,\delta_{n-1}\) are introduced. For dealing with such an extended problem, the stopping times depend on the waiting times values. The theory for such problems is developed.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
62L15 Optimal stopping in statistics
60J60 Diffusion processes
91B70 Stochastic models in economics
91G10 Portfolio theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Beibel M., Statistica Sinica 7 pp 93– (1997)
[2] DOI: 10.1007/s00780-010-0134-8 · Zbl 1303.91167
[3] DOI: 10.1137/09077076X · Zbl 1270.91090
[4] DOI: 10.1239/aap/1158684999 · Zbl 1114.60033
[5] DOI: 10.1007/978-3-0348-8163-0
[6] DOI: 10.1287/moor.1070.0301 · Zbl 1221.60061
[7] DOI: 10.1111/j.1467-9965.2007.00331.x · Zbl 1133.91499
[8] Chow Y. S., Great Expectations: The Theory of Optimal Stopping (1971) · Zbl 0233.60044
[9] DOI: 10.1287/mnsc.1040.0240 · Zbl 1232.90340
[10] DOI: 10.1007/978-1-4684-0302-2
[11] DOI: 10.1214/10-AAP727 · Zbl 1235.60040
[12] DOI: 10.1111/j.0960-1627.2004.00205.x · Zbl 1169.91372
[13] Peskir G., Optimal Stopping and Free-Boundary Problems (2006) · Zbl 1115.60001
[14] DOI: 10.2307/2331121
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.