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Some remarks on \(A(t,B_ t)\). (English) Zbl 0793.60087

Azéma, J. (ed.) et al., Séminaire de probabilités XXVII. Berlin: Springer-Verlag. Lect. Notes Math. 1557, 173-176 (1993).
The main result of this article is the following decomposition \[ A(t,B_ t)=\int^ t_ 0L(s,B_ s)dB_ s+\lim_{\varepsilon\to 0+}{1\over 2}\int^ t_ 0{L(s,B_ s)-L(s,B_ s- \varepsilon)\over\varepsilon}ds. \] Here \(B_ s\) is a standard Brownian motion, \(L(t,x)\) is its local time and \(A(t,x)=\int^ x_{- \infty}L(t,y)dy\). The proof uses the approximation arguments for \[ A(t,B_ t)=\int^ t_ 0I_{\{B_ t-B_ s\geq 0\}}ds=\lim_{\varepsilon\to 0+}\int^ t_ 0\psi_ \varepsilon(B_ t-B_ s)ds, \] where \(\psi_ \varepsilon(x)=\int^ x_ 0\Phi_ \varepsilon(y)dy\), \(\Phi_ \varepsilon\) being an approximate identity. So, it is natural, that the main formula can be interpreted in terms of Schwartz distribution.
For the entire collection see [Zbl 0780.00013].

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
60G48 Generalizations of martingales