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<p>In this paper, we study a class of distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13662_2025_4026_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo>&lt;</mo> <mi>H</mi> <mo>&lt;</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$0&lt; H&lt;1/2$</EquationSource> </InlineEquation> and standard Brownian motion. We first establish the existence and uniqueness theorem for solutions of the considered equations under some non-Lipschitz conditions by Carathéodory approximation technique. Subsequently, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of mean square convergence under certain averaging conditions.</p>

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Distribution dependent stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年11月13日

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  • Ziyan Xie,
  • Zhi Li,
  • Liping Xu

摘要

In this paper, we study a class of distribution dependent stochastic differential equations driven simultaneously by fractional Brownian motion with Hurst 0 < H < 1 / 2 $0< H<1/2$ and standard Brownian motion. We first establish the existence and uniqueness theorem for solutions of the considered equations under some non-Lipschitz conditions by Carathéodory approximation technique. Subsequently, we show that the solutions of distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations in the sense of mean square convergence under certain averaging conditions.