Skip to main content

2019 | OriginalPaper | Buchkapitel

5. Optimal Stopping Rules. General Theory for the Continuous-Time Case

verfasst von : Albert N. Shiryaev

Erschienen in: Stochastic Disorder Problems

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

1. We assume that there are given a filtered probability space \((\Omega , \mathcal {F}, (\mathcal {F}_t)_{t \geqslant 0}, \mathrm {P})\) and a family of stochastic processes \(H =(H_t)_{t \geqslant 0}\), where H t will be interpreted as the payoff if the stopping occurs at the time t.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
17.
Zurück zum Zitat Chow, Y. S., Robbins, H., Siegmund, D. The Theory of Optimal Stopping. Corrected reprint of the 1971 original. Dover Publications, Inc., New York, 1991. Chow, Y. S., Robbins, H., Siegmund, D. The Theory of Optimal Stopping. Corrected reprint of the 1971 original. Dover Publications, Inc., New York, 1991.
20.
Zurück zum Zitat Dellacherie, C., Meyer, P.-A. Probabilités et Potentiel. Chapitres V a VIII. Théorie des Martingales. Revised edition. Actualités Scientifiques et Industrielles, 1385, Herman, Paris, 1980; English translation: Probabilities and potential. B. Theory of martingales. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982. Dellacherie, C., Meyer, P.-A. Probabilités et Potentiel. Chapitres V a VIII. Théorie des Martingales. Revised edition. Actualités Scientifiques et Industrielles, 1385, Herman, Paris, 1980; English translation: Probabilities and potential. B. Theory of martingales. North-Holland Mathematics Studies, 72. North-Holland Publishing Co., Amsterdam, 1982.
22.
Zurück zum Zitat Dubins, L. E., Shepp, L. A., Shiryaev, A. N. Optimal stopping rules and maximal inequalities for Bessel processes. (Russian) Teor. Veroyatnost. i Primenen. 38 (1993), no. 2, 288–330; translation in Theory Probab. Appl. 38 (1993), no. 2, 226–261.MathSciNetCrossRef Dubins, L. E., Shepp, L. A., Shiryaev, A. N. Optimal stopping rules and maximal inequalities for Bessel processes. (Russian) Teor. Veroyatnost. i Primenen. 38 (1993), no. 2, 288–330; translation in Theory Probab. Appl. 38 (1993), no. 2, 226–261.MathSciNetCrossRef
23.
Zurück zum Zitat Dynkin, E. B. Some limit theorems for sums of independent random quantities with infinite mathematical expectations. (Russian) Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 247–266; translation in Select. Transl. Math. Statist. and Probability, Vol. 1 (1961), pp. 171–189 Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I. Dynkin, E. B. Some limit theorems for sums of independent random quantities with infinite mathematical expectations. (Russian) Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 247–266; translation in Select. Transl. Math. Statist. and Probability, Vol. 1 (1961), pp. 171–189 Inst. Math. Statist. and Amer. Math. Soc., Providence, R.I.
24.
Zurück zum Zitat Dynkin, E. B. Markov Processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Dynkin, E. B. Markov Processes. Vols. I, II. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121, 122 Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
35.
Zurück zum Zitat Grigelionis, B. I., Shiryaev, A. N. On the Stefan problem and optimal stopping rules for Markov processes. (Russian) Teor. Veroyatnost. i Primenen. 11 (1966) 612–631; translation in Theor. Probability Appl. 11 (1966), 541–558. Grigelionis, B. I., Shiryaev, A. N. On the Stefan problem and optimal stopping rules for Markov processes. (Russian) Teor. Veroyatnost. i Primenen. 11 (1966) 612–631; translation in Theor. Probability Appl. 11 (1966), 541–558.
44.
Zurück zum Zitat Karatzas, I., Shreve, S. E. Brownian Motion and Stochastic Calculus. 2nd edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. Karatzas, I., Shreve, S. E. Brownian Motion and Stochastic Calculus. 2nd edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991.
48.
Zurück zum Zitat Liptser, R. S., Shiryaev, A. N. Statistics of Random Processes. Nonlinear Filtering and Related Problems. (Russian), Nauka, Moscow, 1974; translated as Statistics of Random processes. I. General Theory, and II. Applications, 2nd revised and expanded edition. Applications of Mathematics (New York), vols. 5 and 6, Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2001. Liptser, R. S., Shiryaev, A. N. Statistics of Random Processes. Nonlinear Filtering and Related Problems. (Russian), Nauka, Moscow, 1974; translated as Statistics of Random processes. I. General Theory, and II. Applications, 2nd revised and expanded edition. Applications of Mathematics (New York), vols. 5 and 6, Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2001.
49.
Zurück zum Zitat Liptser, R. S., Shiryaev, A. N. Martingale Theory. (Russian), Nauka, Moscow, 1986; translated as Theory of Martingales. Mathematics and its Applications (Soviet Series), 49. Kluwer Academic Publishers Group, Dordrecht, 1989. Liptser, R. S., Shiryaev, A. N. Martingale Theory. (Russian), Nauka, Moscow, 1986; translated as Theory of Martingales. Mathematics and its Applications (Soviet Series), 49. Kluwer Academic Publishers Group, Dordrecht, 1989.
55.
Zurück zum Zitat Meyer, P.-A. Probability and Potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. Meyer, P.-A. Probability and Potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966.
66.
Zurück zum Zitat Peskir, G., Shiryaev, A. N. Optimal Stopping and Free-boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2006. Peskir, G., Shiryaev, A. N. Optimal Stopping and Free-boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2006.
76.
Zurück zum Zitat Shewhart, W. A. Economic Control of Quality of Manufactured Products. D. Van Nostrand Co., New York, 1931. Shewhart, W. A. Economic Control of Quality of Manufactured Products. D. Van Nostrand Co., New York, 1931.
77.
Zurück zum Zitat Shewhart, W. A. Statistical Method from the Viewpoint of Quality Control. Second edition. Edited and with a foreword by W. Edwards Deming. Dover Publications, Inc., New York, 1986. (See also Sequential Analysis 26 (2007), no. 2, “Celebrating Eighty Years of Control Charts. Walter A. Shewhart’s Legacy”.) Shewhart, W. A. Statistical Method from the Viewpoint of Quality Control. Second edition. Edited and with a foreword by W. Edwards Deming. Dover Publications, Inc., New York, 1986. (See also Sequential Analysis 26 (2007), no. 2, “Celebrating Eighty Years of Control Charts. Walter A. Shewhart’s Legacy”.)
88.
Zurück zum Zitat Shiryaev, A. N. Statistical Sequential Analysis. Optimal stopping rules. 2nd edition, revised. Izdat. Nauka, Moscow, 1976; 1st edition translated as Statistical Sequential Analysis. Optimal Stopping Rules. Translations of Mathematical Monographs, Vol. 38. American Mathematical Society, Providence, R.I., 1973. Shiryaev, A. N. Statistical Sequential Analysis. Optimal stopping rules. 2nd edition, revised. Izdat. Nauka, Moscow, 1976; 1st edition translated as Statistical Sequential Analysis. Optimal Stopping Rules. Translations of Mathematical Monographs, Vol. 38. American Mathematical Society, Providence, R.I., 1973.
90.
Zurück zum Zitat Shiryaev, A. N. Nonstandard stochastic optimization problems: reduction to problems with Markov representation and their solutions. (Russian) in Contemporary Problems of Mathematics and Mechanics IV no. 1, pp. 8–39 Izd. Mosk. Gosud. Univ., Moscow, 2009. Shiryaev, A. N. Nonstandard stochastic optimization problems: reduction to problems with Markov representation and their solutions. (Russian) in Contemporary Problems of Mathematics and Mechanics IV no. 1, pp. 8–39 Izd. Mosk. Gosud. Univ., Moscow, 2009.
91.
Zurück zum Zitat Shiryaev, A. N. Probabilistic Statistical Methods in Decision Theory. (Russian), Yandex, Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, 2011. Shiryaev, A. N. Probabilistic Statistical Methods in Decision Theory. (Russian), Yandex, Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, 2011.
92.
Zurück zum Zitat Shiryaev, A. N. Probability-1. (Russian), Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2011. Shiryaev, A. N. Probability-1. (Russian), Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2011.
93.
Zurück zum Zitat Shiryaev, A. N. Probability-2. (Russian), Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2011. Shiryaev, A. N. Probability-2. (Russian), Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2011.
94.
Zurück zum Zitat Shiryaev, A. N. Foundations of Stochastic Financial Mathematics Vol.1 (Facts. Models). Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2016. Shiryaev, A. N. Foundations of Stochastic Financial Mathematics Vol.1 (Facts. Models). Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2016.
95.
Zurück zum Zitat Shiryaev, A. N. Foundations of Stochastic Financial Mathematics, Vol.2 (Theory). Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2016. [See also the book Essentials of Stochastic Finance. Facts, Models, Theory. Translated from the Russian manuscript by N. Kruzhilin. Advanced Series on Statistical Science and Applied Probability, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1999.] Shiryaev, A. N. Foundations of Stochastic Financial Mathematics, Vol.2 (Theory). Moscow Center for Continuous Mathematical Education, Independent Univ. of Moscow, Moscow, 2016. [See also the book Essentials of Stochastic Finance. Facts, Models, Theory. Translated from the Russian manuscript by N. Kruzhilin. Advanced Series on Statistical Science and Applied Probability, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1999.]
96.
Zurück zum Zitat Shiryaev, A. N. On martingale methods in the boundary crossing problems for Brownian motion. (Russian) Sovrem. Probl. Mat., 8, Steklov Math. Inst., RAS, Moscow, 2007. Shiryaev, A. N. On martingale methods in the boundary crossing problems for Brownian motion. (Russian) Sovrem. Probl. Mat., 8, Steklov Math. Inst., RAS, Moscow, 2007.
108.
Zurück zum Zitat Wald, A. Sequential Analysis. John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1947. Wald, A. Sequential Analysis. John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1947.
110.
Zurück zum Zitat Zhitlukhin, M. V. Sequential Testing Methods of statistical hypotheses and detection of disorder. (Russian), Dissertation, Steklov Mathematical Institute of the Russian Academy of Sciences, 2013. Zhitlukhin, M. V. Sequential Testing Methods of statistical hypotheses and detection of disorder. (Russian), Dissertation, Steklov Mathematical Institute of the Russian Academy of Sciences, 2013.
115.
Zurück zum Zitat Zhitlukhin, M. V., Shiryaev, A. N. On the existence of solutions of unbounded optimal stopping problems. (Russian) Proc. Steklov Inst. Math. 287 (2014), no. 1, 299–307.MathSciNetCrossRef Zhitlukhin, M. V., Shiryaev, A. N. On the existence of solutions of unbounded optimal stopping problems. (Russian) Proc. Steklov Inst. Math. 287 (2014), no. 1, 299–307.MathSciNetCrossRef
Metadaten
Titel
Optimal Stopping Rules. General Theory for the Continuous-Time Case
verfasst von
Albert N. Shiryaev
Copyright-Jahr
2019
DOI
https://doi.org/10.1007/978-3-030-01526-8_5

Neuer Inhalt