Advances in Decision Sciences
Volume 2006 (2006), Issue 3, Article ID 19181, 24 pages
doi:10.1155/JAMDS/2006/19181

Fundamental solutions to Kolmogorov equations via reduction to canonical form

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.