Summary
Ifs(t, x) is the local time of a Brownian motion, we show the processx→(t, x) is a semi-martingale inH p for allp<∞, with respect to the appropriate excursion fields. In addition, the canonical decomposition of local time, as the sum of a martingale and a process of bounded variation, is found.
Article PDF
Similar content being viewed by others
References
Barlow, M.T.: On Brownian Local Time. In: Séminaire de Probabilités XV, p. 189–190, Lect. Notes in Math.850, Berlin-Heidelberg-New York: Springer 1981
Chung, K.L., Durrett, R.: Downcrossings and local time. Z. Wahrscheinlichkeitstheorie Verw. Gebiete35, 147–149 (1976)
Fisk, D.L.: Quasi-martingales. Trans. Amer. Math. Soc.120, 369–389 (1965)
Hunt, G.: Martingales et Processus de Markov. Paris: Dunod 1966
Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc.109, 56–86 (1963)
McKean, H.P.: Stochastic Integrals. New York: Academic Press 1969.
Ray, D.B.: Sojourn times of diffusion processes. Ill. J. Math. 7, 615–630 (1963)
Walsh, J.B.: Downcrossings and the Markov property of local time. In: Temps Locaux, Astérisque 52–53, pp. 89–115 (1978)
Walsh, J.B.: Excursions and local time. In: Temps Locaux, Astérisque 52–53, pp. 159–192 (1978)
Williams, D.: Lévy's downcrossing theorem. Z. Wahrscheinlichkeitstheorie Verw. Gebiete40, 157–158 (1977)
Williams, D.: Conditional excursion theory. In: Séminaire de Probabilités XIII, pp. 490–494, Lect. Notes in Math.721, Berlin-Heidelberg-New York: Springer 1979
Author information
Authors and Affiliations
Additional information
The author gratefully acknowledges the support of a University Research Fellowship from NSERC of Canada
Rights and permissions
About this article
Cite this article
Perkins, E. Local time is a semi-martingale. Z. Wahrscheinlichkeitstheorie verw Gebiete 60, 79–117 (1982). https://doi.org/10.1007/BF01957098
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01957098