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Temps local et théorie du grossissement, application de la théorie du grossissement à l’étude des temps locaux browniens. (French) Zbl 0562.60080
Grossissements de filtrations: exemples et applications, Sémin. de Calcul stochastique, Paris 1982/83, Lect. Notes Math. 1118, 197-304 (1985).
[For the entire collection see Zbl 0547.00034.]
For a real Brownian motion X its local time \(L^ a_ t\) at a up to \(t\geq 0\) is given by Tanaka’s well known formula \((X_ t-a)_+=(- a)_++\int^{t}_{0}I\{X_ s\geq a\}dX_ s+L^ a_ t.\) Continuing work of D. Williams, Bull. Am. Math. Soc. 75, 1035-1036 (1969; Zbl 0266.60060), the author and M. Yor, Temps locaux. Exposées du séminaire J. Azema-M. Yor (1976-1977). Astérisque 52-53, 1-223 (1978; Zbl 0385.60063), P. McGill, Séminaire de probabilités XV, Univ. Strasbourg 1979/80, Lect. Notes Math. 850, 206-209 (1981; Zbl 0458.60071) and E. Perkins, Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 79-117 (1982; Zbl 0468.60070), the author studies \(L^ a_ t\) and related processes by means of excursion times, changes of the time scale and squares of Bessel processes. For suitable excursion or life times T, \(L^ a_ T\) is a diffusion in \(a\in R\) and its decomposition under enlarged \(\sigma\)-fields is discussed. The paper contains a fascinating collection of delicate results obtained by the techniques of stochastic calculus.
Reviewer: R.J.Elliott

MSC:
60J55 Local time and additive functionals
60H20 Stochastic integral equations
60J65 Brownian motion