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

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.

24 pages

Cited by in corpus (6)

• On the product formula and convolution associated with the index Whittaker transform
• Markov processes related to the stationary measure for the open KPZ equation
• Markov limits of steady states of the KPZ equation on an interval
• Lévy processes with respect to the index Whittaker convolution
• Sturm-Liouville hypergroups without the compactness axiom
• Localization POVMs and intrinsic temporal uncertainty