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≥0} of the form
E₀ [ f(B_t)g (∫^t₀ φ(B_s)ds) ]
have the asymptotics t^-3/2 as t→∞ for some suitable non-negative functions φ, 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.