Rough path integral of local time

Feng, Chunrong; Zhao, Huaizhong

doi:10.1016/j.crma.2008.02.015

Probability Theory

Rough path integral of local time
[Intégrale curviligne non régulière du temps local]

Présenté par : Marc Yor

Feng, Chunrong ^1,^2,³ ; Zhao, Huaizhong ¹

¹ Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
² School of Mathematics and System Sciences, Shandong University, Jinan, 250100, China
³ Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China

Comptes Rendus. Mathématique, Tome 346 (2008) no. 7-8, pp. 431-434.

Résumé
Abstract

Dans cette Note, pour un temps local d'une semi-martingale continue, nous définissons l'intégrale $\int_{- \infty}^{\infty} g (x) d L_{t}^{x}$ pour toute fonction g de q-variation finie ( $2 ⩽ q < 3$ ) en utilisant l'intégrale de Lyons pour des chemins non-réguliers. Nous obtenons alors la formule de Tanaka–Meyer pour une fonction continue f lorsque $\nabla^{-} f$ existe et est de q-variation finie avec $2 ⩽ q < 3$ . Le cas correspondant à $1 ⩽ q < 2$ utilise l'intégrale de Young (voir Feng et Zhao [C.R. Feng, H.Z. Zhao, Two-parameter $p, q$ -variation path and integration of local times, Potential Analysis 25 (2006) 165–204.]).

In this Note, for a continuous semimartingale local time $L_{t}^{x}$ , we establish the integral $\int_{- \infty}^{\infty} g (x) d L_{t}^{x}$ as a rough path integral for any finite q-variation function g ( $2 ⩽ q < 3$ ) by using Lyons' rough path integration. We therefore obtain the Tanaka–Meyer formula for a continuous function f if $\nabla^{-} f$ exists and is of finite q-variation, $2 ⩽ q < 3$ . The case when $1 ⩽ q < 2$ was established by Feng and Zhao [C.R. Feng, H.Z. Zhao, Two-parameter $p, q$ -variation path and integration of local times, Potential Analysis 25 (2006) 165–204] using the Young integral.

Reçu le : 2006-12-01
Accepté le : 2008-02-12
Publié le : 2008-03-28

DOI : 10.1016/j.crma.2008.02.015

Affiliations des auteurs :

Feng, Chunrong ^{1, 2, 3} ; Zhao, Huaizhong ¹

@article{CRMATH_2008__346_7-8_431_0,
     author = {Feng, Chunrong and Zhao, Huaizhong},
     title = {Rough path integral of local time},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {431--434},
     publisher = {Elsevier},
     volume = {346},
     number = {7-8},
     year = {2008},
     doi = {10.1016/j.crma.2008.02.015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.02.015/}
}

TY  - JOUR
AU  - Feng, Chunrong
AU  - Zhao, Huaizhong
TI  - Rough path integral of local time
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 431
EP  - 434
VL  - 346
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.02.015/
DO  - 10.1016/j.crma.2008.02.015
LA  - en
ID  - CRMATH_2008__346_7-8_431_0
ER  -

%0 Journal Article
%A Feng, Chunrong
%A Zhao, Huaizhong
%T Rough path integral of local time
%J Comptes Rendus. Mathématique
%D 2008
%P 431-434
%V 346
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.02.015/
%R 10.1016/j.crma.2008.02.015
%G en
%F CRMATH_2008__346_7-8_431_0

Feng, Chunrong; Zhao, Huaizhong. Rough path integral of local time. Comptes Rendus. Mathématique, Tome 346 (2008) no. 7-8, pp. 431-434. doi : 10.1016/j.crma.2008.02.015. http://www.numdam.org/articles/10.1016/j.crma.2008.02.015/

Bibliographie
Cité par

[1] Feng, C.R.; Zhao, H.Z. Two-parameter $p, q$ -variation path and integration of local times, Potential Analysis, Volume 25 (2006), pp. 165-204

[2] Lejay, A. An introduction to rough paths, Sèminaire de probabilitès XXXVII, Lecture Notes in Mathematics, vol. 1832, Springer-Verlag, 2003, pp. 1-59

[3] Lyons, T.; Qian, Z. System Control and Rough Paths, Clarendon Press, Oxford, 2002

[4] Karatzas, I.; Shreve, S.E. Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1998

[5] Revuz, D.; Yor, M. Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin, 1994

Cité par Sources :