CAUCHY PROBLEM FOR ULTRA-PARABOLIC EQUATIONS OF KOLMOGOROV TYPE WITH BLOCK STRUCTURE

Abstract

The investigation is devoted to ultra-parabolic equations which appear in Asian options problems. Unlike the European option, the payout of Asian derivative depends on the entire trajectory of the price value, not the final value only. Among methods of researching of the Asian options, the one is to include dependent on the price trajectory variables in the state space. The expansion of the state space by including of dependent on the price trajectory variables transforms the path-dependent problem for the Asian option into an equivalent path-independent Markov problem. However, the increasing of the dimension usually leads to partial differential equations which are not uniformly parabolic. The class of these equations under some conditions is a generalization of the well-known degenerate parabolic A.N.Kolmogorov's equation of diffusion with inertia. Mathematical models of the options have been studied in many works. It has been constructed so called L-type fundamental solutions for considered equations previously, some their properties have been established, the Cauchy problem has been researched. In current work, for the given equations we study the classical solutions of the Cauchy problem. For the coefficients of the equations we apply special Hölder conditions with respect to spatial variables. Under these conditions, we prove the wellposedness of the Cauchy problem in special weighed spaces, obtained integral presentation of classic solutions of the Cauchy problem for homogeneous equations. Classes of well-posedness of the Cauchy problem were described. The results obtained in the work are realization of well-known Eidelman-Ivasyshen approach. Ones can be used to advanced studying of the Cauchy problem and boundary value problems for linear and quasi-linear degenerated parabolic equations, as well as in the theory of stochastic processes when studying Markov processes, the transition probability density of which is the fundamental solution of the Cauchy problem for these equations.