# The spectral expansion approach to index transforms and connections with the theory of diffusion processes

## TL;DR

This paper explores how spectral theory of Sturm-Liouville operators underpins index transforms, linking them to diffusion processes and offering new tools for mathematical finance applications.

## Contribution

It establishes a general connection between index transforms, spectral theory, and diffusion processes, extending the Yor integral concept and applying Feynman-Kac theorem.

## Key findings

- Extended Yor integral to Sturm-Liouville transforms
- Connected index transforms with diffusion processes via spectral theory
- Demonstrated applications in mathematical finance

## Abstract

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations. We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.