Probability Theory
Rough path integral of local time
Comptes Rendus. Mathématique, Volume 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.]).

Accepted:
Published online:
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, Volume 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/

[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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