On the distribution of integral functionals of a homogeneous diffusion process

M Perestyuk,G Shevchenko,Yu Mishura

doi:10.15559/vmsta-2014.10

Abstract

In this article, we study homogeneous transient diffusion processes. We provide the basic distributions of their local times. It helps to get exact formulas and upper bounds for the moments, exponential moments, and potentials of integral functionals of transient diffusion processes. Some of the results generalize the corresponding results of Salminen and Yor for the Brownian motion with drift.

Highlights

We consider a family {Xtx, t ≥ 0, x ∈ R} of one-dimensional homogeneous diffusion processes defined on a complete filtered probability space {Ω, F, {Ft }t≥0, P} by a stochastic differential equation dXtx = b Xtx dt + a Xtx dWt, t ≥ 0, (1)with initial condition X0x = x ∈ R, where {Wt, t ≥ 0} is a standard Ft -Wiener process
We further introduce several objects related to the family {Xtx, t ≥ 0, x ∈ R}
For any (a, b) ⊂ R and x ∈ (a, b), let τax,b = inf{t ≥ 0, Xtx ∈/ (a, b)} = τax ∧ τbx be the first moment of exiting the interval (a, b). (We use the convention inf ∅ = +∞.)

Summary

Introduction

For any (a, b) ⊂ R and x ∈ (a, b), let τax,b = inf{t ≥ 0, Xtx ∈/ (a, b)} = τax ∧ τbx be the first moment of exiting the interval (a, b). For any t > 0 and y ∈ R, define the local time of the process Xx at the point y on the interval [0, t] by. (The factor a2(y) is included to agree with the general Meyer–Tanaka definition of the local time of a semimartingale [7].) The limit in (2) exists almost surely and defines a continuous nondecreasing process {Lxt (y), t ≥ 0} for any x, y ∈ R. We follow the approach of Salminen and Yor [8] to study integral functionals of a Wiener process with positive drift and generalize their results to homogeneous transient diffusion processes. Applying the results of [6], we establish criteria of convergence of almost sure finiteness of the functionals J∞(f ), calculate their moments and potentials, and bound their exponential moments

The distribution of the local time of a transient diffusion process

Integral functionals of a transient diffusion processes