Bessel bridges decomposition with varying dimension. Applications to finance

TL;DR

This paper extends classical Bessel process theory to processes with time-varying dimensions, providing new decompositions, computational methods, and financial applications for these generalized stochastic models.

Contribution

It introduces a novel decomposition of Bessel bridges with time-dependent dimensions and develops methods to compute their Laplace transforms, with applications in finance.

Findings

Decomposition of Bessel bridges with variable dimension

Methodology for Laplace transform computation of additive functionals

Financial applications demonstrating model versatility

Abstract

We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results.