<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the <italic>L<sup>p</sup></italic> and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.</p>
</abstract>

The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the Lp and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.

An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not...

An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Published Online: Nov 23, 2021

Page range: 105 - 133

Received: Mar 22, 2021

Accepted: Apr 30, 2021

DOI: https://doi.org/10.2478/auom-2021-0037

Keywords
Fréchet derivative, and almost sure convergence, Polynomial condition, Stochastic differential equations, Taylor approximation

© 2021 Dušan D. Djordjević et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Published Online: Nov 23, 2021

Page range: 105 - 133

Received: Mar 22, 2021

Accepted: Apr 30, 2021

DOI: https://doi.org/10.2478/auom-2021-0037

KeywordsFréchet derivative, and almost sure convergence, Polynomial condition, Stochastic differential equations, Taylor approximation

© 2021 Dušan D. Djordjević et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Fréchet derivative, and almost sure convergence, Polynomial condition, Stochastic differential equations, Taylor approximation