The study is devoted to ultraparabolic equations that arise in problems describing Asian options in financial markets. The class of such equations under certain conditions is a generalization of the well-known degenerate parabolic diffusion equation with inertia of A.M. Kolmogorov. Previously, for equations from this class, the so-called fundamental L-solution was constructed, its properties and correct solvability of the Cauchy problem were investigated.

This work formulates special Hölder conditions for spatial variables on the coefficients of such equations, under which the correct solvability of the Cauchy problem in special weight spaces is obtained, as well as integral representations of classical solutions of homogeneous equations in the form of Poisson integrals of functions or generalized measures, which specify the initial condition. The correctness classes of the Cauchy problem are described.

The obtained results can be used in further studies of the Cauchy problem and boundary value problems for linear and quasilinear degenerate parabolic equations, as well as in the study of Markov processes, the transition probability density of which is the FRZK for the equations under consideration.

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The study is devoted to ultraparabolic equations that arise in problems describing Asian options in financial markets. The class of such equations under certain conditions is a generalization of the well-known degenerate parabolic diffusion equation with inertia of A.M. Kolmogorov. Previously, for equations from this class, the so-called fundamental L-solution was constructed, its properties and correct solvability of the Cauchy problem were investigated.[4] Barraquand J., Pudet T. Pricing of American path-dependent contingent claims. Math. Finance 1996, 6, 17–51.
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