fundamental solution of the Cauchy problem

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 E$_{22}^{B}$.  This class is a generalization of the well-known class of degenerate parabolic Kolmogorov type equations E$_{22}$.  So called $L$-type fundamental solutions have been constructed for the equations from the class E$_{22}^{B}$ previously, and some their properties have been established as well.  The main feature of the

Some properties of the fundamental solution of the Cauchy problem for homogeneous ultraparabolic Kolmogorov–type equation with three groups of spatial variables including two groups of degeneration and with degeneration on the initial hyperplane are established. For different type of degeneration on the initial hyperplane the theorems on integral representations of solutions and correct solvability of the Cauchy problem are presented.  These results for such type of equations are obtained in appropriate classes of weight functions.