The investigation is devoted to ultra-parabolic equations with two group of spatial variables 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. Among the main problems in the study of the Asian options models when they are reduced to ultra-parabolic equations of the Kolmogorov type there are the following: the construction, researching of the existence, uniqueness and properties (for instance, such as non-negativity, normality, convolution formula) of the fundamental solution of the Cauchy problem as the probability density of the transition between the states of the stochastic process, which given by the corresponding stochastic differential equation.

It has been constructed so called $L$-type fundamental solutions for equations from the class previously, and some their properties have been established. In the work, it is formulated some known results about $L$-type fundamental solutions.

In current research, for the equations from this class we build and study the classical fundamental solutions of the Cauchy problem. For the coefficients of the equations we apply special H\"older conditions with respect to spatial variables. We prove the existing of the classic fundamental solutions and its properties such as estimates, including estimates of the derivatives, normality, convolution formula, positivity etc.

The results obtained in the work can be used to receive the well-posedness of the Cauchy problem for such equations in the classical sense.

[1] Barraquand J., Pudet T. Pricing of American path-dependent contingent claims. Math. Finance 1996,
6, 17–51.
[2] Black F., Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 1973,
81, 637–659.
[3] Di Francesco, Pascucci A. On a class of degenerate parabolic equations of Kolmogorov type. AMRX
Appl. Math. Res. Exprass 2005, 3, 77–116.
[4] Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and
pseudo-differential equations of parabolic type. Birkh¨auser. Basel, 2004, Ser. Operator Theory: Adv.
and Appl., Vol. 152. doi: 10.1007/978-3-0348-7844-9.
[5] Foschi P., Pascucci A. Kolmogorov equations arising in finance: direct and inverse problem. Lecture
Notes of Seminario Interdisciplinare di Matematica. Universit´a degli Studi della Basilicata 2007, VI,
145–156.
[6] Frentz M., Nystr¨om K., Pascucci A., Polidoro S. Optimal regularity in the obstacle problem for
Kolmogorov operators related to American Asian options. Math. Ann. 2010, 347, 805–838. doi:
10.1007/s00208-009-0456-z
[7] Ivashyshen S.D., Layuk V.V. The fundamental solutions of the Cauchy problem for some degenerate
parabolic equations of Kolmogorov type Ukr. Mat. J. 2011, 63 (11), 1469–1500 (in Ukrainian).
[8] Ivasyshen S.D., Medyns’kyi I.P. Classical fundamental solution of the Cauchy problem for ultraparabolic
Kolmogorov-type equations with two qroups of spatial variables of degeneration. I. J. Math. Sci. 2020,
246 (2), 121–151. doi: 10.1007/s10958-020-04726-z
[9] Ivasyshen S.D., Medyns’kyi I.P. Classical fundamental solution of the Cauchy problem for ultraparabolic
Kolmogorov-type equations with two qroups of spatial variables of degeneration. II. J. Math. Sci. 2020,
247 (1), 1–23. doi: 10.1007/s10958-020-04786-z
[10] Ivasyshen S.D., Medyns’kyi I.P. On classical fundamental solutions of the Cauchy problem for
ultraparabolic Kolmogorov-type equations with two qroups of spatial variables. Mat. Metody i Fiz.-Mekh.
Polya 2016, 59 (2), 28–42 (in Ukrainian).
[11] Kolmogorov A. Zufllige Bewegungen. (Zur Theorie der Brownschen Bewegung.). Ann. of Math., II. Ser.
1934, 35, 116–117.
[12] Mishura Yu.S., Ralchenko K.V., Sakhno M.L., Shevchenko G.M. Stochastic processes: theory, statistics,
application: textbook. 2nd Edition. Kyiv University, Kyiv, 2023 (in Ukrainian).
[13] Pascucci A. Free boundary and optimal stopping problems for American Asian options. Finance and
Stoch. 2008, 12, 21–41. doi: 10.1007/s00780-007-0051-7
[14] Polidoro S. On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Le Matematiche
1994, 49 (1), 53–105.
[15] Stanton R. Path Dependent Payoffs and Contingent Claim Valuation: Single Premium Deferred Annuities.
Unpublished manuscript, Graduate School of Business, Stanford University, 1989.