In this paper, a new approach is derived to approximate solutions of stochastic differential equations by using stochastic analysis on time scales. We will prove that the solutions of stochastic dynamic equations (SDEs) on a sequence of time scales <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1898_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\mathbb {T}_n\}_{n=1}^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi mathvariant="double-struck">T</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> converge in mean-square to the solution of SDEs on time scale <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1898_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation> provided that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1898_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {T}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">T</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> tends to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1898_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation> in Hausdorff topology. Further, the rate of convergence can be estimated if the drift and diffusion coefficients are Lipschitz in both variables t and x. This work can be considered as a generalization of the Euler-Maruyama approximation for solving numerical solutions of stochastic differential equations.

A New Approach to Approximate Solutions of Stochastic Differential Equations

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发表日期：

2025年5月24日

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Do Duc Thuan,
Nguyen Thanh Dieu,
Nguyen Huu Du

摘要

In this paper, a new approach is derived to approximate solutions of stochastic differential equations by using stochastic analysis on time scales. We will prove that the solutions of stochastic dynamic equations (SDEs) on a sequence of time scales $\{\mathbb {T}_n\}_{n=1}^\infty $ ${T_{n}}_{n = 1}^{\infty}$ converge in mean-square to the solution of SDEs on time scale $\mathbb {T}$ $T$ provided that $\mathbb {T}_n$ $T_{n}$ tends to $\mathbb {T}$ $T$ in Hausdorff topology. Further, the rate of convergence can be estimated if the drift and diffusion coefficients are Lipschitz in both variables t and x. This work can be considered as a generalization of the Euler-Maruyama approximation for solving numerical solutions of stochastic differential equations.