Rough path integral of local time

Chunrong Feng; Huaizhong Zhao

doi:10.1016/j.crma.2008.02.015

Probability Theory

Rough path integral of local time

Presented by: Marc Yor

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

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

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

Abstract (Original language)
Résumé

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.

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.]).

Received: 2006-12-01
Accepted: 2008-02-12
Published online: 2008-03-28

DOI: 10.1016/j.crma.2008.02.015

Author's affiliations:

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

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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

References
Cited by

[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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