Abstract
Let B be a one-dimensional Brownian motion and \(f: {\mathbb{R}}{\longrightarrow}{\mathbb{R}}\) be a Borel function that is locally integrable on \(\mathbb{R}\backslash \{0\}\). We present necessary and sufficient conditions (in terms of the function f) for the existence of the limit
in probability and almost surely. This limit (if it exists) can be called the principal value of the integral \(\int_0^tf(B_s)ds\).
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 zero energy.
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© 2001 Springer-Verlag Berlin/Heidelberg
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Cherny, A.S. (2001). Principal Values of the Integral Functionals of Brownian Motion: Existence, Continuity and an Extension of Itô’s Formula. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXV. Lecture Notes in Mathematics, vol 1755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44671-2_24
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DOI: https://doi.org/10.1007/978-3-540-44671-2_24
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