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Principal values of the integral functionals of Brownian motion: Existence, continuity and an extension of Itô’s formula. (English) Zbl 0977.60075

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 348-370 (2001).
Summary: Let \(B\) be a one-dimensional Brownian motion and \(f:\mathbb{R}\to \mathbb{R}\) be a Borel function that is locally integrable on \(\mathbb{R}\setminus\{0\}\). We present necessary and sufficient conditions (in terms of the function \(f\)) for the existence of the limit \(\lim_{\varepsilon\downarrow 0} \int^t_0 f(B_s) I(|B_s|> \varepsilon) ds\) in probability and almost surely. This limit (if it exists) can be called the principal value of the integral \(\int^t_0 f(B_s) ds\). The obtained results are applied to give an extension of Itô’s formula with the principal value as the covariation term. We also show that the principal value defines a continuous additive functional of zero energy.
For the entire collection see [Zbl 0960.00020].

MSC:

60J65 Brownian motion
60H05 Stochastic integrals
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