Hideaki Uemura

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(Bt−x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and Bt the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.

Substitution to parametrized generalized Wiener functionals and pinned Brownian local times

Abstract We discuss how to substitute Wiener functionals for parameters in generalized Wiener functionals: Let {Φ(x);x∈ R N } be generalized Wiener functionals parametrized by x∈ R N and F be a non-degenerate Wiener functional in Malliavins sense. Then, formally, it holds that Φ(F)=∫ R N Φ(x)δ x (F) d x . We define the substitution by justification of the formal representation above. Under this definition we show the relationship between pinned Brownian local times and Brownian local times.