A new construction of the <i>σ</i>-finite measures associated with submartingales of class (<i>Σ</i>)

Joseph Najnudel; Ashkan Nikeghbali

doi:10.1016/j.crma.2010.01.021

Probability Theory

A new construction of the σ-finite measures associated with submartingales of class (Σ)
[Une nouvelle construction des mesures σ-finies associées aux sous-martingales de classe (Σ)]

Présenté par : Marc Yor

Joseph Najnudel ¹ ; Ashkan Nikeghbali ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 311-316.

Résumé
Abstract

Dans Najnudel et Nikeghbali (2009) [7], nous prouvons que pour toute sous-martingale ${(X_{t})}_{t ⩾ 0}$ de classe (Σ), définie sur un espace de probabilité filtré $(Ω, F, P, {(F_{t})}_{t ⩾ 0})$ , satisfaisant certaines conditions techniques, on peut construire une mesure σ-finie $Q$ sur $(Ω, F)$ , telle que pour tout $t ⩾ 0$ , et pour tout événement $Λ_{t} \in F_{t}$ :

Q [Λ_{t}, g ⩽ t] = E_{P} [1_{Λ_{t}} X_{t}]

où g est le dernier zéro de X. Certains cas particuliers de cette construction sont liés aux pénalisations browniennes ou aux mathématiques financières. Dans cette note, nous donnons une construction plus simple de

Q

, et nous montrons qu'un analogue de cette mesure peut aussi être défini pour des sous-martingales à temps discret.

In Najnudel and Nikeghbali (2009) [7], we prove that for any submartingale ${(X_{t})}_{t ⩾ 0}$ of class (Σ), defined on a filtered probability space $(Ω, F, P, {(F_{t})}_{t ⩾ 0})$ , which satisfies some technical conditions, one can construct a σ-finite measure $Q$ on $(Ω, F)$ , such that for all $t ⩾ 0$ , and for all events $Λ_{t} \in F_{t}$ :

Q [Λ_{t}, g ⩽ t] = E_{P} [1_{Λ_{t}} X_{t}]

where g is the last hitting time of zero of the process X. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this Note, we give a simpler construction of

Q

, and we show that an analog of this measure can also be defined for discrete-time submartingales.

Reçu le : 2009-12-23
Accepté le : 2010-01-12
Publié le : 2010-02-26

DOI : 10.1016/j.crma.2010.01.021

Affiliations des auteurs :

Joseph Najnudel ¹ ; Ashkan Nikeghbali ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

@article{CRMATH_2010__348_5-6_311_0,
     author = {Joseph Najnudel and Ashkan Nikeghbali},
     title = {A new construction of the \protect\emph{\ensuremath{\sigma}}-finite measures associated with submartingales of class {(\protect\emph{\ensuremath{\Sigma}})}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {311--316},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.021},
     language = {en},
}

TY  - JOUR
AU  - Joseph Najnudel
AU  - Ashkan Nikeghbali
TI  - A new construction of the σ-finite measures associated with submartingales of class (Σ)
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 311
EP  - 316
VL  - 348
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.021
LA  - en
ID  - CRMATH_2010__348_5-6_311_0
ER  -

%0 Journal Article
%A Joseph Najnudel
%A Ashkan Nikeghbali
%T A new construction of the σ-finite measures associated with submartingales of class (Σ)
%J Comptes Rendus. Mathématique
%D 2010
%P 311-316
%V 348
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2010.01.021
%G en
%F CRMATH_2010__348_5-6_311_0

Joseph Najnudel; Ashkan Nikeghbali. A new construction of the σ-finite measures associated with submartingales of class (Σ). Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 311-316. doi : 10.1016/j.crma.2010.01.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.021/

Bibliographie
Cité par

[1] A. Bentata, M. Yor, From Black–Scholes and Dupire formulae to last passage times of local martingales. Part A: The infinite time horizon, 2008

[2] K. Bichteler Stochastic Integration and Stochastic Differential Equations, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2002

[3] P. Cheridito; A. Nikeghbali; E. Platen Processes of the class sigma, last zero and draw-down processes, 2009 | arXiv

[4] D. Madan, B. Roynette, M. Yor, From Black–Scholes formula, to local times and last passage times for certain submartingales, Prépublication IECN 2008/14

[5] J. Najnudel; A. Nikeghbali A new kind of augmentation of filtrations, 2009 | arXiv

[6] J. Najnudel; A. Nikeghbali On some properties of a universal sigma-finite measure associated with a remarkable class of submartingales, 2009 | arXiv

[7] J. Najnudel; A. Nikeghbali On some universal σ-finite measures and some extensions of Doob's optional stopping theorem, 2009 | arXiv

[8] J. Najnudel; B. Roynette; M. Yor A global view of Brownian penalisations, MSJ Memoirs, vol. 19, Mathematical Society of Japan, Tokyo, 2009

[9] A. Nikeghbali A class of remarkable submartingales, Stochastic Process. Appl., Volume 116 (2006) no. 6, pp. 917-938

[10] K.-R. Parthasarathy Probability Measures on Metric Spaces, Academic Press, New York, 1967

[11] C. Profeta; B. Roynette; M. Yor Option Prices as Probabilities: A New Look at Generalized Black–Scholes Formulae, Springer Finance, 2010

[12] D.-W. Stroock; S.-R.-S. Varadhan Multidimensional Diffusion Processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006 (reprint of the 1997 edition)

[13] M. Yor Les inégalités de sous-martingales, comme conséquences de la relation de domination, Stochastics, Volume 3 (1979) no. 1, pp. 1-15

Fulgence Eyi Obiang; Octave Moutsinga; Youssef Ouknine An ideal class to construct solutions for skew Brownian motion equations, Journal of Theoretical Probability, Volume 35 (2022) no. 2, pp. 894-916 | DOI:10.1007/s10959-021-01078-5 | Zbl:1502.60037
Yun Li; Benedek Valkó Operator level limit of the circular Jacobi $β$ -ensemble, Random Matrices: Theory and Applications, Volume 11 (2022) no. 4, p. 41 (Id/No 2250043) | DOI:10.1142/s2010326322500435 | Zbl:1504.60014
Fulgence Eyi-Obiang; Youssef Ouknine; Octave Moutsinga On the study of processes of $\sum (H)$ and $\sum_{s} (H)$ classes, Journal of Theoretical Probability, Volume 30 (2017) no. 1, pp. 117-142 | DOI:10.1007/s10959-015-0640-x | Zbl:1378.60069
Fulgence Eyi-Obiang; Youssef Ouknine; Octave Moutsinga; Gerald Trutnau Some contributions to the study of stochastic processes of the classes $Σ (H)$ and $(Σ)$ , Stochastics, Volume 89 (2017) no. 8, pp. 1253-1269 | DOI:10.1080/17442508.2017.1307984 | Zbl:1394.60056
Joseph Najnudel On $σ$ -finite measures related to the Martin boundary of recurrent Markov chains, Séminaire de Probabilités XLVII, Cham: Springer, 2015, pp. 263-297 | DOI:10.1007/978-3-319-18585-9_13 | Zbl:1336.60144
Paul-Olivier Dehaye; Dirk Zeindler On averages of randomized class functions on the symmetric groups and their asymptotics, Annales de l'Institut Fourier, Volume 63 (2013) no. 4, pp. 1227-1262 | DOI:10.5802/aif.2802 | Zbl:1308.60017
Christopher Hughes; Joseph Najnudel; Ashkan Nikeghbali; Dirk Zeindler Random permutation matrices under the generalized Ewens measure, The Annals of Applied Probability, Volume 23 (2013) no. 3, pp. 987-1024 | DOI:10.1214/12-aap862 | Zbl:1276.60009
Joseph Najnudel; Ashkan Nikeghbali On some universal $σ$ -finite measures related to a remarkable class of submartingales, Stochastic Processes and their Applications, Volume 122 (2012) no. 4, pp. 1582-1600 | DOI:10.1016/j.spa.2012.01.010 | Zbl:1275.60045
Monique Jeanblanc; Shiqi Song Random times with given survival probability and their $F$ -martingale decomposition formula, Stochastic Processes and their Applications, Volume 121 (2011) no. 6, pp. 1389-1410 | DOI:10.1016/j.spa.2011.03.003 | Zbl:1230.60047

Cité par 9 documents. Sources : zbMATH

Commentaires - Politique

Il n'y a aucun commentaire pour cet article. Soyez le premier à écrire un commentaire !

Publier un nouveau commentaire:

Publier une nouvelle réponse: