Electronic Journal of Probability

Convergence of values in optimal stopping and convergence of optimal stopping times

François Coquet and Sandrine Toldo

Full-text: Open access

Abstract

Under the hypothesis of convergence in probability of a sequence of càdlàg processes $(X^n)$ to a càdlàg process $X$, we are interested in the convergence of corresponding values in optimal stopping and also in the convergence of optimal stopping times. We give results under hypothesis of inclusion of filtrations or convergence of filtrations.

Article information

Source
Electron. J. Probab. Volume 12 (2007), paper no. 8, 207-228.

Dates
Accepted: 27 February 2007
First available in Project Euclid: 1 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1464818479

Digital Object Identifier
doi:10.1214/EJP.v12-288

Mathematical Reviews number (MathSciNet)
MR2299917

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 62L15 60Fxx 62Lxx: Sequential methods

Keywords
Values in optimal stopping Convergence of stochastic processes Convergence of filtrations Optimal stopping times Convergence of stopping times

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Coquet, François; Toldo, Sandrine. Convergence of values in optimal stopping and convergence of optimal stopping times. Electron. J. Probab. 12 (2007), paper no. 8, 207--228. doi:10.1214/EJP.v12-288. http://projecteuclid.org/euclid.ejp/1464818479.


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References

  • Aldous, David. Stopping times and tightness. Ann. Probability 6 (1978), no. 2, 335–340.
  • D. Aldous. Weak convergence of stochastic processes for processes viewed in the Strasbourg manner. Unpublished Manuscript, Statis. Laboratory Univ. Cambridge, 1981.
  • Aldous, David. Stopping times and tightness. II. Ann. Probab. 17 (1989), no. 2, 586–595.
  • Amin, Kaushik; Khanna, Ajay. Convergence of American option values from discrete- to continuous-time financial models. Math. Finance 4 (1994), no. 4, 289–304.
  • Baxter, J. R.; Chacon, R. V. Compactness of stopping times. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), no. 3, 169–181.
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9
  • Brémaud, Pierre; Yor, Marc. Changes of filtrations and of probability measures. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 4, 269–295.
  • Coquet, François; Mémin, Jean; Słominski, Leszek. On weak convergence of filtrations. Séminaire de Probabilités, XXXV, 306–328, Lecture Notes in Math., 1755, Springer, Berlin, 2001.
  • Hoover, D. N. Convergence in distribution and Skorokhod convergence for the general theory of processes. Probab. Theory Related Fields 89 (1991), no. 3, 239–259.
  • Itô, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp.
  • Jacod, Jean; Mémin, Jean. Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité. (French) [A type of intermediate convergence between convergence in law and convergence in probability] Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 529–546, Lecture Notes in Math., 850, Springer, Berlin-New York, 1981.
  • Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 1987. xviii+601 pp. ISBN: 3-540-17882-1
  • El Karoui, N. Les aspects probabilistes du contrôle stochastique. (French) [The probabilistic aspects of stochastic control] Ninth Saint Flour Probability Summer School–-1979 (Saint Flour, 1979), pp. 73–238, Lecture Notes in Math., 876, Springer, Berlin-New York, 1981.
  • Knight, Frank B. On the random walk and Brownian motion. Trans. Amer. Math. Soc. 103 1962 218–228.
  • D. Lamberton. Convergence of the critical price in the approximation of american options. Math. Finance, 3 (2):179–190, 1993.
  • Lamberton, Damien; Lapeyre, Bernard. Introduction au calcul stochastique appliqué à la finance. (French) [Introduction to stochastic calculus applied to finance] Second edition. Ellipses, Édition Marketing, Paris, 1997. 176 pp. ISBN: 2-7298-4782-0.
  • Lamberton, Damien; Pagès, Gilles. Sur l'approximation des réduites. (French) [On the approximation of reduites] Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), no. 2, 331–355.
  • Mémin, Jean. Stability of Doob-Meyer decomposition under extended convergence. Acta Math. Appl. Sin. Engl. Ser. 19 (2003), no. 2, 177–190.
  • Meyer, P.-A. Convergence faible et compacité des temps d'arrêt d'après Baxter et Chacon. (French) Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977), pp. 411–423, Lecture Notes in Math., 649, Springer, Berlin, 1978.
  • Mulinacci, Sabrina; Pratelli, Maurizio. Functional convergence of Snell envelopes: applications to American options approximations. Finance Stoch. 2 (1998), no. 3, 311–327.
  • Rényi, Alfréd. On stable sequences of events. Sankhyā Ser. A 25 1963 293 302.
  • Shiryayev, A. N. Optimal stopping rules. Translated from the Russian by A. B. Aries. Applications of Mathematics, Vol. 8. Springer-Verlag, New York-Heidelberg, 1978. x+217 pp. ISBN: 0-387-90256-2
  • Shiryaev, Albert N. Essentials of stochastic finance. Facts, models, theory. Translated from the Russian manuscript by N. Kruzhilin. Advanced Series on Statistical Science & Applied Probability, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1999. xvi+834 pp. ISBN: 981-02-3605-0