Analytic approach to Yor’s formula of exponential additive functionals of Brownian motion

Shin-ichi Kotani
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Shin-ichi Kotani: Osaka University, Department of Mathematics, Graduate School of Science

A chapter in Itô’s Stochastic Calculus and Probability Theory, 1996, pp 185-195 from Springer

Abstract: Abstract Yor [4] obtained an exact formula for a one-dimensional Brownian motion {B t }: $${{E}_{0}}(f(\int_{0}^{t}{{{e}^{2{{B}_{S}}}}d}s)g({{e}^{{{B}_{t}}}}))=c(t)\int_{0}^{\infty }{dzg(y)f(\frac{1}{z})}\exp \left\{ -\frac{z(1+{{y}^{2}})}{2}{{\psi }_{yz}}(t) \right\},$$ $$c(t)={{(2{{\pi }^{3}}t)}^{-\frac{1}{2}}}\exp (\frac{{{\pi }^{2}}}{2t}),{{\psi }_{r}}(t)=\int_{0}^{\infty }{\exp }(-\frac{{{u}^{2}}}{2t}-\gamma \cosh u)\sinh u\sin (\frac{\pi u}{t})du.$$

Keywords: Green Function; Heat Kernel; Neumann Boundary Condition; Additive Functional; Tauberian Theorem (search for similar items in EconPapers)
Date: 1996
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DOI: 10.1007/978-4-431-68532-6_12

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