Abstract

We investigate the transition density of a hyperbolic Bessel process for integer dimensions and show a link between transition densities of a hyperbolic Brownian motion and a Bessel process in the same dimension. Using the so-called Millson’s formula for the densities of hyperbolic Brownian motion we also show a link between transition density of $n$-dimensional hyperbolic Bessel process and $2$-dimensional (if $n$ is even) or $3$-dimensional (if $n$ is odd) hyperbolic Brownian motion. This helps us to get explicit formulas for the Bessel process transition density.

Highlights

  • Hyperbolic Bessel process is one of many diffusion processes that are used to model different physical and economic phenomena (e.g. Asian options pricing, see [11])

  • We investigate the transition density of a hyperbolic Bessel process for integer dimensions and show a link between transition densities of a hyperbolic Brownian motion and a Bessel process in the same dimension

  • Using the so-called Millson’s formula for the densities of hyperbolic Brownian motion we show a link between transition density of n-dimensional hyperbolic Bessel process and 2-dimensional or 3-dimensional hyperbolic Brownian motion

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Summary

Introduction

Hyperbolic Bessel process is one of many diffusion processes that are used to model different physical and economic phenomena (e.g. Asian options pricing, see [11]) This process has been recently investigated by Jakubowski and Wisniewolski in a paper [9], earlier its properties were investigated by Gruet in [6], [7], [8] and by Borodin in [1]. Transition density of a hyperbolic Bessel process for investigation of stable processes (and more general: subordinated Brownian motions) with values in hyperbolic spaces. Such investigations have just started, compare for instance [13]. In this paper we obtain such manageable formulas for the transition densities of hyperbolic Bessel processes of integer dimension, that is for the radial parts of hyperbolic Brownian motions. In order to avoid misunderstanding, all densities we consider in this paper, if taken with respect to the Lebesgue measure, will be denoted by p (with sub- or superscripts), otherwise they will be denoted with bar: p

Hyperbolic spaces and hyperbolic processes
Hyperbolic space
Hyperbolic Brownian motion and its density
Hyperbolic Bessel process of integer dimension
Transition density of the hyperbolic Bessel process
Examples

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