A time-change approach to Kotani's extension of Yor's formula

Abstract

In [3], Kotani proved analytically that expectations for additive functionals of Brownian motion $\{B_t,t\ge 0\}$ of the form $$E_0\left[f\left(B_t\right)g\left({\int }_0^t\phi \left(B_s\right)ds\right)\right]$$ have the asymptotics $t^{-3/2}$ as $t\to \mathrm{\infty }$ for some suitable non-negative functions $\phi$, $f$ and $g$. This generalizes, in the asymptotic form, Yor's explicit formula [10] 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.