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<p>This study introduces three numerical techniques to obtain the solutions of stochastic evolution equations driven by infinite-dimensional fractional Brownian motion. The fractional Brownian motion, which is defined by a Hurst parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41980_2024_937_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\in \left( \frac{1}{2},1\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, is a Gaussian process widely applied in modeling various natural and engineered systems. We use the Galerkin method for spatial discretization with three distinct time discretization schemes to approximate the solutions. We specifically consider <i>Q</i>-fractional Brownian motion, where <i>Q</i> represents a trace class operator. This paper seeks to establish theoretical results on the convergence and error bounds of the proposed methods, supplemented by validation through numerical experiments.</p>

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Convergence of Three Numerical Approaches for Stochastic Evolution Equations with Fractional Brownian Motion

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Minoo Kamrani

摘要

This study introduces three numerical techniques to obtain the solutions of stochastic evolution equations driven by infinite-dimensional fractional Brownian motion. The fractional Brownian motion, which is defined by a Hurst parameter \(H\in \left( \frac{1}{2},1\right) \) H ∈ 1 2 , 1 , is a Gaussian process widely applied in modeling various natural and engineered systems. We use the Galerkin method for spatial discretization with three distinct time discretization schemes to approximate the solutions. We specifically consider Q-fractional Brownian motion, where Q represents a trace class operator. This paper seeks to establish theoretical results on the convergence and error bounds of the proposed methods, supplemented by validation through numerical experiments.