On fundamental solution of the Cauchy problem for ultra-parabolic equations in the Asian options models

Abstract

Paper studies ultra-parabolic equations with three groups of spatial variables appearing in Asian options problems. The class of these equations which satisfy some conditions was denoted by EB22. This class is a generalization of the well-known class of degenerate parabolic Kolmogorov type equations E22. So called L-type fundamental solutions have been constructed for the equations from the class EB22 previously, and some their properties have been established as well. The main feature of the research was the establishing of an one-to-one correspondence between the classes EB22 and E22. The Cauchy problem classic fundamental solutions for the equations from the class EB22 are considered. Special H\"older conditions with respect to spatial variables are applied to the coefficients of the equations.