no. 7-8
Probability Theory
Rough path integral of local time
[Intégrale curviligne non régulière du temps local]
Présenté par : Marc Yor
Feng, Chunrong1, 2, 3 ; Zhao, Huaizhong1
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

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.

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.]).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.02.015

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 
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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

[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

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