On hyperbolic Bessel processes and beyond

TL;DR

This paper explores the properties of hyperbolic Bessel processes, providing explicit formulas, probabilistic representations, and applications such as a new proof of Bougerol's identity, advancing understanding of these stochastic processes.

Contribution

It introduces explicit Laplace transform formulas and distributional representations for hyperbolic Bessel processes, including new proofs and generalizations of known identities.

Findings

Derived explicit Laplace transform formulas.

Expressed distributions in terms of known processes.

Provided new proofs of Bougerol's identity.

Abstract

We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosine of hyperbolic Bessel processes and functionals of geometric Brownian motion. We present an explicit formula for the Laplace transform of the hyperbolic cosine of a hyperbolic Bessel process and some other interesting probabilistic representations of this Laplace transform. We express the one-dimensional distribution of a hyperbolic Bessel process in terms of other, known and independent processes. We present some applications including a new proof of Bougerol's identity and its generalization. We characterize the distribution of the process which is the hyperbolic sine of hyperbolic Bessel process.