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<p>Driven by the emergence of stochastic differential equations (SDEs) influenced by <i>G</i>-Brownian motion in the context of uncertain data across various financial scenarios, there is a pressing need to develop efficient numerical schemes for approximating these types of SDEs. Recently, several discretization schemes have been introduced to numerically solve <i>G</i>-SDEs using the standard Euler–Maruyama method. This study presents a first-order discretization scheme based on the <i>G</i>-Itô’s formula for <i>G</i>-SDEs. Furthermore, we examine the convergence of the proposed scheme and explore its asymptotic behavior in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10959_2025_1441_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> sense. The findings confirm that the numerical method exhibits stability against small perturbations.</p>

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Milstein Scheme for Stochastic Differential Equations driven by G-Brownian Motion

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Bahar Akhtari,
  • Panyu Wu

摘要

Driven by the emergence of stochastic differential equations (SDEs) influenced by G-Brownian motion in the context of uncertain data across various financial scenarios, there is a pressing need to develop efficient numerical schemes for approximating these types of SDEs. Recently, several discretization schemes have been introduced to numerically solve G-SDEs using the standard Euler–Maruyama method. This study presents a first-order discretization scheme based on the G-Itô’s formula for G-SDEs. Furthermore, we examine the convergence of the proposed scheme and explore its asymptotic behavior in the \(L^2\) L 2 sense. The findings confirm that the numerical method exhibits stability against small perturbations.