High-order approximation of Pearson diffusion processes

Leonenko, Ganna M. ORCID: https://orcid.org/0000-0001-8025-661X and Phillips, Timothy Nigel ORCID: https://orcid.org/0000-0001-6455-1205 2012. High-order approximation of Pearson diffusion processes. Journal of Computational and Applied Mathematics 236 (11) , pp. 2853-2868. 10.1016/j.cam.2012.01.022

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Abstract

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics Medicine MRC Centre for Neuropsychiatric Genetics and Genomics (CNGG)
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Pearson diffusions; Reduced basis; Spectral approximation
Publisher:	Elsevier
ISSN:	0377-0427
Last Modified:	18 Oct 2022 14:30
URI:	https://orca.cardiff.ac.uk/id/eprint/17786

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