# American Institute of Mathematical Sciences

November  2018, 17(6): 2351-2378. doi: 10.3934/cpaa.2018112

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

 CMUP, Departamento de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

* Corresponding author

Received  June 2017 Revised  February 2018 Published  June 2018

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.

Citation: Rúben Sousa, Semyon Yakubovich. The spectral expansion approach to index transforms and connections with the theory of diffusion processes. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2351-2378. doi: 10.3934/cpaa.2018112
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