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On a quasi everywhere existence of the local time of the 1-dimensional Brownian motion. (English) Zbl 0551.60076

The one-dimensional Brownian motion can itself be considered as the state of an Ornstein-Uhlenbeck process with state-space the space of continuous functions. Thus one can perform a stochastic variation of Brownian motion. It is natural to ask which properties of the Brownian motion path still hold good along the stochastic variation through the Ornstein- Uhlenbeck process. This paper shows that the local time can be defined so as to vary continuously with the stochastic variation. The method is to apply Kolmogorov’s continuity criterion to the stochastic integral occurring in Tanaka’s formula for local time.
Reviewer: Wilfrid S.Kendall

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
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