# GREEN'S FUNCTION OF ONE CLASS OF SECOND-ORDER DEGENERATE PARABOLIC EQUATIONS

## DOI:

https://doi.org/10.31471/2304-7399-2022-17(64)-44-57## Keywords:

Green's function, fundamental solution, Kolmogorov's equation, Lévy's method, degenerate parabolic equations, diffusion processes.## Abstract

**Abstract**. The Green's function of a linear degenerate parabolic equation, which has 4n degrees of freedom, generalizes the Kolmogorov equation, is constructed in the article. The coefficients of the equation are continuous, bounded and satisfy the Hölder condition with the index 0<α≤1. The explicit form of the fundamental solution of the equation with parameters is used. The Levy method is applied to the properties of the fundamental solution of high-order Kolmogorov-type equations with coefficients dependent only on t, in particular, the point that is the parametric is selected so that the exponential estimation of the fundamental solution and its derivatives is conveniently used. The existence and properties of the fundamental solutions of the Kolmogorov equations with coefficients depending on the time variable, as well as the behavior of the volume potential generated by the parametrics, were investigated. Estimates of the derivatives of the fundamental solution have been established. The interest in diffusion equations with inertia is caused by their application in economic theory, in particular in the theory of options, and their wide application in the study of derivative pricing processes. Analytical formulas reflecting the Green's function are consistent with empirical data and, when applied in practice, adequately reflect the course of processes on the stock markets. This presentation allows you to calculate the market value of a portfolio of shares, assess internal volatility in the market at any time, and also analyze the dynamics of the stock market.

## References

Weber M. The fundamental solution of degenerate partial differential equation of parabolic type. Trans. Amer. Math. Soc. 1951, 1 (71), 24-37

Eidelman S. D., Ivasyshen S. D., Malytska H. P. A modified Levi method: development and application // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. – 1998. – № 5. – P. 14–19.

Malytska A., Burtnyak, І.V. On the Fundamental Solution of the Cauchy Problem for Kolmogorov Systems of the Second Order. Ukrainian Mathematical Journal. Volume 70, Issue 8, 1 January 2019, 1275-1287. DOI: 10.1007/s11253-018-1568-y.

Eidelman S. D. Parabolic systems. – Amsterdam: North-Holland Pub. Co., 1969. – 475 p.

Friedman A. Partial differential equations of parabolic type. – Englewood Cliffs: Prentice-Hall, 1964. – xiv+347 p.

Eidelman S. D., Ivasyshen S. D., Kochubei A. N. Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. – Basel etc.: Birkh¨auser, 2004. – IX + 387 p.

Polidoro S. On a class of ultraparabolic operators of Kolmogorov–Fokker–Planck type // Le Mathematiche. – 1994. – 49. – P. 53–105.

Burtnyak, І.V. Malytska A. Application of the spectral theory and perturbation theory to the study of Ornstein-Uhlenbesck processes. Carpathian Math. Publ. 2018, 10 (2), 273–287. doi:10.15330/cmp.10.2.273-287.

Lorig M.J. Pricing Derivatives on Multiscale Diffusions: an Eigenfunction Expansion Approach. Math. Finance 2014, 24 (2), 331–363.

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*PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number*, (17(64), 44–57. https://doi.org/10.31471/2304-7399-2022-17(64)-44-57