<p>In this paper, we study a nonparametric estimation of random effects from the following distribution dependent SDE driven by fractional Brownian motions <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_Equa.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="470" /> </MediaObject> <EquationSource Format="TEX">\(\begin{aligned} dX^j_t=\beta _jb(t,X^j_t,\mathcal {L}_{X_t^j})dt+\varepsilon \sigma (t, \mathcal {L}_{X^j_t})dB_t^{j,H}, ~X^j_0=x^j_0, ~0\le t\le T, \end{aligned}\)</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">\(j=1,2,\cdots,N\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq2.gif" Format="GIF" Height="24" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {L}_{X^j_t}\)</EquationSource> </InlineEquation> denotes the law of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq3.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(X^j_t\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">\(B_t^{1, H},\cdots,B_t^{N, H}\)</EquationSource> </InlineEquation> are independent fractional Brownian motions with common Hurst parameter <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="90" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\in (1/2,1)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta _j\)</EquationSource> </InlineEquation> is random variable independent of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(B^{j,H}\)</EquationSource> </InlineEquation>. This result extends the existing results of non parametric estimation to the case of distribution dependent. Moreover, we consider the estimation for the density function of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="362_2025_1742_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\beta _j\)</EquationSource> </InlineEquation> under weaker conditions of kernel function and study the corresponding asymptotic distribution.</p>

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Nonparametric estimation for distribution dependent SDEs driven by fractional brownian motions with random effects

  • Guangjun Shen,
  • Qian Yu,
  • Huan Zhou

摘要

In this paper, we study a nonparametric estimation of random effects from the following distribution dependent SDE driven by fractional Brownian motions \(\begin{aligned} dX^j_t=\beta _jb(t,X^j_t,\mathcal {L}_{X_t^j})dt+\varepsilon \sigma (t, \mathcal {L}_{X^j_t})dB_t^{j,H}, ~X^j_0=x^j_0, ~0\le t\le T, \end{aligned}\) where \(j=1,2,\cdots,N\) , \(\mathcal {L}_{X^j_t}\) denotes the law of \(X^j_t\) , \(B_t^{1, H},\cdots,B_t^{N, H}\) are independent fractional Brownian motions with common Hurst parameter \(H\in (1/2,1)\) and \(\beta _j\) is random variable independent of \(B^{j,H}\) . This result extends the existing results of non parametric estimation to the case of distribution dependent. Moreover, we consider the estimation for the density function of \(\beta _j\) under weaker conditions of kernel function and study the corresponding asymptotic distribution.