×

zbMATH — the first resource for mathematics

Optimal multiple stopping time problem. (English) Zbl 1235.60040
The authors show that computing the value function for the \(d\)-time optimal stopping problem \[ v(S)= \text{ess\,sup}\{\operatorname{E}[\psi(\tau_1, \tau_2,\dots, \tau_d)|{\mathcal F}_S],\,\tau_1,\dots, \tau_d\in T_S\} \] reduces to computing the value function for an optimal single stopping time problem \[ v(S)= \text{ess\,sup}\{\operatorname{E}[\phi(\theta)|{\mathcal F}_S],\,\theta\in T_S\} \] a.s. for a reward \(\phi\) expressible in terms of optimal \((d-1)\)-stopping time problems, i.e., by induction, the initial optimal \(d\)-stopping time problem can be reduced to nested optimal single stopping time problems. Sufficient regularity of \(\psi\) assumed, \(\phi\) is shown to have a progressively measurable càdlàg modification, which opens a way for interesting applications, e.g., in finance.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G07 General theory of stochastic processes
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
62L15 Optimal stopping in statistics
91G80 Financial applications of other theories
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Carmona, R. and Dayanik, S. (2008). Optimal multiple stopping of linear diffusions. Math. Oper. Res. 33 446-460. · Zbl 1221.60061
[2] Carmona, R. and Touzi, N. (2008). Optimal multiple stopping and valuation of swing options. Math. Finance 18 239-268. · Zbl 1133.91499
[3] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel , Chap. I-IV , nouvelle édition. Hermann, Paris. · Zbl 0323.60039
[4] El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. In École d’été de Probabilités de Saint-Flour IX- 1979. Lect. Notes in Math. 876 73-238. Springer, Berlin. · Zbl 0472.60002
[5] Karatzas, I. and Shreve, S. E. (1994). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0734.60060
[6] Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Applications of Mathematics ( New York ) 39 . Springer, New York. · Zbl 0941.91032
[7] Kobylanski, M. and Quenez, M. C. (2010). Optimal multiple stopping in the Markovian case and applications to finance. Working paper. · Zbl 1189.60086
[8] Kobylanski, M., Quenez, M.-C. and Rouy-Mironescu, E. (2010). Optimal double stopping time problem. C. R. Math. Acad. Sci. Paris 348 65-69. · Zbl 1189.60086
[9] Kobylanski, M. and Rouy, E. (1998). Large deviations estimates for diffusion processes with Lipschitz reflections. Thèse de Doctorat de L’université de Tours de M. Kobylanski 17-62.
[10] Maingueneau, M. A. (1978). Temps d’arrêt optimaux et théorie générale. In Séminaire de Probabilités , XII ( Univ. Strasbourg , Strasbourg , 1976 / 1977). Lecture Notes in Math. 649 457-467. Springer, Berlin. · Zbl 0376.60050
[11] Neveu, J. (1975). Discrete-Parameter Martingales , revised ed. North-Holland Mathematical Library 10 . North-Holland, Amsterdam. Translated from the French by T. P. Speed. · Zbl 0345.60026
[12] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems . Birkhäuser, Basel. · Zbl 1115.60001
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.