Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Azéma, J., Gundy, R. F. and Yor, M.: Sur l’intégrabilité uniforme des martin-gales continues. In Seminar on Probability, XIV (Paris, 1978/1979) (French), volume 784 of Lecture Notes in Math., pages 53-61. Springer, Berlin, 1980.
Azéma, J. and Yor, M.: Le problème de Skorokhod: compléments à “Une solution simple au problème de Skorokhod”. In Séminaire de Probabilités, XIII, volume 721 of Lecture Notes in Math., pages 625-633. Springer, Berlin, 1979.
Azéma, J. and Yor, M.: Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII, volume 721 of Lecture Notes in Math., pages 90-115. Springer, Berlin, 1979.
Blackwell, D. and Dubins, L.E.: A converse to the dominated convergence theorem. Illinois J. Math., 7:508-514, 1963.
Dellacherie, C. and Weil, M. (editors): Séminaire de Probabilités. XIII, vol-ume 721 of Lecture Notes in Mathematics. Springer, Berlin, 1979. Held at the Université de Strasbourg, Strasbourg, 1977/78.
Dubins, L. E.: Personal communication with M. Yor. 2004.
L. E. Dubins and D. Gilat. On the distribution of maxima of martingales. Proc. Amer. Math. Soc., 68(3):337-338, 1978.
Dubins, L. E. and Schwarz, G.: A sharp inequality for sub-martingales and stopping-times. Astérisque, (157-158):129-145, 1988. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987).
Dubins, L. E., Shepp, L. A. and Shiryaev, A. N.: Optimal stopping rules and maximal inequalities for Bessel processes. Teor. Veroyatnost. i Primenen., 38(2):288-330, 1993.
Elworthy, K. D., Li, X. M. and Yor, M.: On the tails of the supremum and the quadratic variation of strictly local martingales. In Séminaire de Probabilités, XXXI, volume 1655 of Lecture Notes in Math., pages 113-125. Springer, Berlin, 1997.
Fernandez-Ponce, J. M., Kochar, S. C. and Muñoz-Perez, J.: Partial orderings of distributions based on right-spread functions. J. Appl. Probab., 35(1):221-228, 1998.
Garsia, A. M.: Martingale Inequalities: Seminar Notes on Recent Progress. W.A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. Mathematics Lecture Notes Series.
Hobson, D. G.: The maximum maximum of a martingale. In Séminaire de Probabilités, XXXII, volume 1686 of Lecture Notes in Math., pages 250-263. Springer, Berlin, 1998.
Kertz, R. P. and Rösler, U.: Martingales with given maxima and terminal distributions. Israel J. Math., 69(2):173-192, 1990.
Kochar, S. C., Li, X. and Shaked, M.: The total time on test transform and the excess wealth stochastic orders of distributions. Adv. in Appl. Probab., 34(4):826-845, 2002.
Lehoczky, J. P.: Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probability, 5(4):601-607, 1977.
Neveu, J.: Discrete-parameter Martingales. North-Holland Publishing Co., Am-sterdam, revised edition, 1975. Translated from the French by T. P. Speed, North-Holland Mathematical Library, Vol. 10.
Oblój, J.: The Skorokhod embedding problem and its offspring. Probability Surveys, 1:321-392, 2004.
Oblój, J.: A complete characterization of local martingales which are functions of Brownian motion and its supremum. Technical Report 984, LPMA - University of Paris 6, 2005. ArXiv: math.PR/0504462.
Oblój, J. and Yor, M.: An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale. Stochastic Process. Appl., 110(1):83-110, 2004.
Peskir, G. and Shiryaev, A. N.: On the Brownian first-passage time over a one-sided stochastic boundary. Teor. Veroyatnost. i Primenen., 42(3):591-602, 1997.
Peskir, G. and Shiryaev, A. N.: Maximal inequalities for reflected Brownian motion with drift. Teor. Imovir. Mat. Stat., (63):125-131, 2000.
Pierre, M: Le problème de Skorokhod: une remarque sur la démonstration d’Azéma-Yor. In Séminaire de Probabilités, XIV (Paris, 1978/1979) (French), volume 784 of Lecture Notes in Math., pages 392-396. Springer, Berlin, 1980.
Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Princi-ples of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999.
Rogers, L. C. G.: The joint law of the maximum and terminal value of a mar-tingale. Probab. Theory Related Fields, 95(4):451-466, 1993.
Roynette, B., Vallois, P. and Yor, M.: Limiting laws associated with brown-ian motion perturbed by its maximum, minimum and local time II. Technical Report 51, Institut Elie Cartan, 2004. to appear in Studia Sci. Math. Hungar.
Roynette, B., Vallois, P. and Yor, M.: Pénalisations et extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère. Techni-cal Report 31, Institut Elie Cartan, 2004. To appear in Séminaire de Probabilités XXXIX, Lecture Notes in Math., Springer, 2005.
Shaked, M. and Shanthikumar, J. G.: Two variability orders. Probab. Engrg. Inform. Sci., 12(1):1-23, 1998.
Shepp, L. and Shiryaev, A. N.: The Russian option: reduced regret. Ann. Appl. Probab., 3(3):631-640, 1993.
Shepp, L. A. and Shiryaev, A. N.: A new look at the “Russian option”. Teor. Veroyatnost. i Primenen., 39(1):130-149, 1994.
L. A. Shepp and A. N. Shiryaev.: The Russian option under conditions of possible “freezing” of prices. Uspekhi Mat. Nauk, 56(1(337)):187-188, 2001.
Tsirelson, B.S.: An example of a stochastic differential equation that has no strong solution. Teor. Verojatnost. i Primenen., 20(2):427-430, 1975.
Vallois, P.: Le problème de Skorokhod sur R: une approche avec le temps local. In Séminaire de Probabilités, XVII, volume 986 of Lecture Notes in Math., pages 227-239. Springer, Berlin, 1983.
Vallois, P.: Quelques inégalités avec le temps local en zero du mouvement brown-ien. Stochastic Process. Appl., 41(1):117-155, 1992.
Vallois, P.: Sur la loi du maximum et du temps local d’une martingale continue uniformement intégrable. Proc. London Math. Soc. (3), 69(2):399-427, 1994.
Williams, D.: Probability with martingales. Cambridge Mathematical Text-books. Cambridge University Press, Cambridge, 1991.
Yor, M.: De nouveaux résultats sur l’équation de Tsirel′son. C. R. Acad. Sci. Paris Sér. I Math., 309(7):511-514, 1989.
Yor, M.: Tsirel′son’s equation in discrete time. Probab. Theory Related Fields, 91(2):135-152, 1992.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Oblój, J., Yor, M. (2006). On Local Martingale and its Supremum: Harmonic Functions and beyond. In: From Stochastic Calculus to Mathematical Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30788-4_25
Download citation
DOI: https://doi.org/10.1007/978-3-540-30788-4_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30782-2
Online ISBN: 978-3-540-30788-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)