A time-change approach to Kotani’s extension of Yor’s formula. (English) Zbl 1093.60052

Hariya, Yuu

J. Math. Soc. Japan 58, No. 1, 129-151 (2006).

Author’s abstract: Kotani proved analytically that expectations for additive functionals of Brownian motion \(\{B_t\), \(t\geq 0\}\) of the form \(E_0 [f(B_t)g(\int_0^t \varphi(B_s)\,ds)]\) have the asymptotics \(t^{-3/2}\) as \(t\to\infty\) for some suitable nonnegative functions \(\varphi,~f\) and \(g.\) This generalizes, in the asymptotic form, Yor’s explicit formula for exponential Brownian functionals. In the present paper, we discuss this generalization probabilistically, by using a time-change argument. We may easily see from our argument that this asymptotics \(t^{-3/2}\) comes from the transition probability of \(3\)-dimensional Bessel process.

Reviewer: Josef Steinebach (Köln)

MSC:

60J65	Brownian motion
60J35	Transition functions, generators and resolvents
60J55	Local time and additive functionals
60F99	Limit theorems in probability theory

Keywords:

additive exponential functional; Brownian motion; transition probability; 3-dimensional Bessel process