Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T20:20:56.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Two chain-transformations and their applications to quantiles

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

J. Bertoin ,
L. Chaumont  and
M. Yor
J. Bertoin*
Affiliation:
Université Pierre et Marie Curie
L. Chaumont*
Affiliation:
Université Pierre et Marie Curie
M. Yor*
Affiliation:
Université Pierre et Marie Curie
*
∗Postal address: Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France.
∗Postal address: Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France.
∗Postal address: Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris Cedex 05, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe two chain-transformations which explain and extend identities for order statistics and quantiles proved by Wendel, Port and, more recently, by Dassios.

Keywords

ORDER STATISTICSQUANTILEEXCHANGEABILITYCHAIN TRANSFORMATION

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 882 - 897
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Akahori, J. (1995) Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.Google Scholar
[2] Sparre Andersen, E. (1953) On sums of symmetrically dependent random variables. Scand. Akt. Tidskr. 26, 123–138.Google Scholar
[3] Bertoin, J. (1991) Décomposition du mouvement brownien avec dérive en un minimum local par juxtaposition de ses excursions positives et négatives. Séminaire de Probabilités XXV. (Lecture Notes in Mathematics 1485.) Springer, Berlin. pp. 330–344,Google Scholar
[4] Bertoin, J. (1993) Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stoch. Proc. Appl. 47, 17–35.Google Scholar
[5] Bingham, N. H. (1975) Fluctuation theory in continuous time. Adv. Appl. Prob. 7, 705–766.Google Scholar
[6] Dassios, A. (1995) The distribution of the quantiles of a Brownian motion with drift and the pricing of related path-dependent options. Ann. Appl. Prob. 5, 389–398.Google Scholar
[7] Dassios, A. (1996) Sample quantiles of stochastic processes with stationary and independent increments and of sums of exchangeable random variables. Ann. Appl. Prob. 6, 1041–1043.Google Scholar
[8] Dassios, A. (1996) Sample quantiles of additive renewal reward processes. J. Appl. Prob. 33, 1018–1032.Google Scholar
[9] Doney, R. A. (1987) On the maxima of random walks and stable processes and the arc-sine law. Bull. London Math. Soc. 19, 177–182.Google Scholar
[10] Doney, R. A. (1991) Hitting probabilities for spectrally positive Lévy processes. J. London Math. Soc. 44, 566–476.Google Scholar
[11] Doney, R. A. (1993) A path decomposition for Lévy processes. Stoch. Proc. Appl. 47, 167–181.CrossRefGoogle Scholar
[12] Embrechts, P., Rogers, L. C. G. and Yor, M. (1995) A proof of Dassios' representation of the a-quantile of Brownian motion with drift. Ann. Appl. Prob. 5, 757–767.Google Scholar
[13] Feller, W. E. (1971) An Introduction to Probability Theory and its Applications. Vol. II. 2nd edn. Wiley, New York.Google Scholar
[14] Knight, F. B. (1996) The uniform law for exchangeable and Lévy process bridges. In: Hommage á P. A. Meyer et J. Neveau. Astérisque 236, 171–187.Google Scholar
[15] Meyer, P. A. (1976) Un cours sur les intégrales stochastiques. Séminaire de Probabilités X. (Lecture Notes in Mathematics 511.) Springer, Berlin, pp. 245–400.Google Scholar
[16] Miura, R. (1992) A note on a look-back option based on order statistics. Hitotsubashi J. Commerce Management 27, 15–28.Google Scholar
[17] Pecherskii, E. A. and Rogozin, B. A. (1969) On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Prob. Appl. 14, 410–423.Google Scholar
[18] Port, S. C. (1963) An elementary approach to fluctuation theory. J. Math. Anal. Appl. 6, 109–151.Google Scholar
[19] Protter, P. (1990) Stochastic Integration and Stochastic Differential Equations: A New Approach. Springer, Berlin.Google Scholar
[20] Wendel, J. G. (1960) Order statistics of partial sums. Ann. Math. Statist. 31, 1034–1044.Google Scholar
[21] Williams, D. (1969) Markov properties of Brownian local time. Bull. Amer. Math. Soc. 75, 1035–1036.Google Scholar
[22] Yor, M. (1978) Rappels et préliminaires généraux. In Temps Locaux, Astérisque 52-53, 17–22.Google Scholar
[23] Yor, ?. (1995) The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.Google Scholar