Principal Values of the Integral Functionals of Brownian Motion: Existence, Continuity and an Extension of Itô’s Formula
نویسنده
چکیده
Let B be a one-dimensional Brownian motion and f : R → R be a Borel function that is locally integrable on R \ {0} . We present necessary and sufficient conditions (in terms of the function f ) for the existence of the limit lim ε↓0 ∫ t 0 f(Bs) I(|Bs| > ε) ds in probability and almost surely. This limit (if it exists) can be called the principal value of the integral ∫ t 0 f(Bs) 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.
منابع مشابه
Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion
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