Potential Analysis

, Volume 13, Issue 4, pp 303–328 | Cite as

Integration with respect to local time

  • Nathalie Eisenbaum


Let \(\left( {L_t^x ;x \in \mathbb{R},t \geqslant 0} \right)\) be the local time process of a linear Brownian motion B. We integrate the Borel functions on \(\mathbb{R}_ \times \mathbb{R}_ + \) with respect to \(\left( {L_t^x ;x \in \mathbb{R},t \geqslant 0} \right)\). This allows us to write Itôrs formula for new classes of functions, and to define a local time process of B on any borelian curve. Some results are extended from deterministic to random functions.

Brownian motion local time stochastic integration Itô formula 


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  1. 1.
    Azéma, J., Jeulin, T., Knight, F. and Yor, M.: ‘Quelques calculs de compensateurs impliquant l'injectivité de certains processus croissants’. Séminaire de Probabilités XXXII (1998), LNM 1686, 316–327.Google Scholar
  2. 2.
    Bouleau, N. and Yor, M.: ‘Sur la variation quadratique des temps locaux de certaines semimartingales’. C.R. Acad. Sc. Paris 292 (1981), 491–494.Google Scholar
  3. 3.
    Dozzi, M.: ‘Stochastic processes with a multidimensional parameter’. Pitman Research Notes in Mathematics Series (1989).Google Scholar
  4. 4.
    Föllmer, H.: ‘Quasimartingales à deux indices’. C.R. Acad. Sci. Paris 288 (1977), 28–49.Google Scholar
  5. 5.
    Föllmer, H., Protter, Ph. and Shiryaev, A. N.: ‘Quadratic covariation and an extension of Itô's formula’. Bernoulli 1(1/2) (1995), 149–169.Google Scholar
  6. 6.
    Kunita, H.: ‘Stochastic flows and Stochastic differential equations’. Cambridge Studies in Advanced Mathematics 24. Cambridge University Press (1990).Google Scholar
  7. 7.
    Lyons, T. J. and Zhang, T. S.: ‘Decomposition of Dirichlet processes and its application’. Annals of Probas 22(1) (1994), 494–524.Google Scholar
  8. 8.
    Lyons, T. J. and Zheng, W. A.: ‘Diffusion processes with non-smooth diffusion coefficients and their density functions’. Proceedings of the Royal Society of Edinburg 115A (1990), 231–242.Google Scholar
  9. 9.
    Meyer, P. A.: ‘Formule d'Itô géneralisée pour le mouvement brownien linéaire, d'aprés Föllmer, Protter, Shiryaev’. Sém de Probas. XXXI, LNM 1655 (1997), 252–255.Google Scholar
  10. 10.
    Pardoux, E. and Protter, P.: ‘A two-sided stochastic integral and its calculus’. Probab. Theory. Relat. Fields 76 (1987), 15–49.Google Scholar
  11. 11.
    Perkins, E.: ‘Local time is a semi-martingale’. Zeitschrift für Wahr 60 (1982), 79–117.Google Scholar
  12. 12.
    Rogers, L. C. G. and Walsh, J. B.: ‘Local time and stochastic area integrals’. Annals of Probas. 19(2) (1991), 457–482.Google Scholar
  13. 13.
    Revuz, D. and Yor, M.: Continuous Martingales and Brownian Motion. Springer-Verlag, 1991.Google Scholar
  14. 14.
    Russo, F. and Vallois, P.: ‘Itô formula for C 1-functions of semimartingales’. Probab. Theory Relat. Fields 104 (1996), 27–41.Google Scholar
  15. 15.
    Stoica, L.: ‘On two-parameter semmartingales’. Zeitschrift für Wahr 45 (1978), 257–268.Google Scholar
  16. 16.
    Sznitman, A. S.: ‘Martingales dépendant d'un paramètre: une formule d'Itô’. Zeitschrift für Wahr 60 (1982), 41–70.Google Scholar
  17. 17.
    Walsh, J. B.: ‘Stochastic integration with respect to local time’, in Cinlar, Chung and Getoor (eds), Seminar on Stochastic Processes, Birkhaüser, Boston, 1982, pp. 237–302.Google Scholar

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© Kluwer Academic Publishers 2000

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  • Nathalie Eisenbaum

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