Comptes Rendus
no. 7-8
Probability Theory
Rough path integral of local time
[Intégrale curviligne non régulière du temps local]

Présenté par : Marc Yor

Chunrong Feng1, 2, 3 ; Huaizhong Zhao1
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, Volume 346 (2008) no. 7-8, pp. 431-434.

Dans cette Note, pour un temps local d'une semi-martingale continue, nous définissons l'intégrale ∫−∞∞g(x)dLtx 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 ∇−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 Ltx, we establish the integral ∫−∞∞g(x)dLtx 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 ∇−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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.02.015

Chunrong Feng 1, 2, 3 ; Huaizhong Zhao 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 
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Chunrong Feng; Huaizhong Zhao. Rough path integral of local time. Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 431-434. doi : 10.1016/j.crma.2008.02.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.02.015/

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

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

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

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

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

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