Journal of Applied Mathematics and Decision Sciences
Volume 2006 (2006), Article ID 19181, 24 pages
doi:10.1155/JAMDS/2006/19181
Fundamental solutions to Kolmogorov equations via reduction to canonical form
School of Mathematics and Applied Statistics, University of Wollongong, Wollongong 2522, NSW, Australia
Received 25 January 2006; Revised 11 May 2006; Accepted 13 June 2006
Copyright © 2006 Joanna Goard. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper finds fundamental solutions to the backward Kolmogorov
equations, usually interpretable as transition density functions
for variables x that follow certain stochastic processes of the
form dx=A(x,t)dt+cxydX and dx=A(x,t)dt+α1+α2x+α3x2dX. This is achieved by first reducing the relevant PDEs
that the density functions satisfy to their canonical form. These
stochastic processes have direct realistic applications in the
modeling of financial assets.