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  4. On fundamental solution of the Cauchy problem for ultra-parabolic equations in the Asian options models

On fundamental solution of the Cauchy problem for ultra-parabolic equations in the Asian options models

MMC.
2024;
Volume 11, Number 2
: pp. 593–606
https://doi.org/10.23939/mmc2024.02.593
Received: October 10, 2023
Revised: June 25, 2024
Accepted: June 29, 2024

Dron V. S., Medynskyi I. P.  On fundamental solution of the Cauchy problem for ultra-parabolic equations in the Asian options models.  Mathematical Modeling and Computing. Vol. 11, No. 2, pp. 593–606 (2024)

Authors:
  1. V. S. Dron
  2. I. P. Medynskyi
1
Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of the National Academy of Sciences of Ukraine
2
Lviv Polytechnic National University

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 research was the establishing of an one-to-one correspondence between the classes E$_{22}^{B}$ and E$_{22}$.  The Cauchy problem classic fundamental solutions for the equations from the class E$_{22}^{B}$ are considered. Special H\"older conditions with respect to spatial variables are applied to the coefficients of the equations.

Asian options
ultra-parabolic equation of Kolmogorov type
fundamental solution of the Cauchy problem
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