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<p>In this paper, we consider a class of stochastic Volterra integral equations with general singular kernels, driven by a Brownian motion and a pure jump Lévy process. We first show that these equations have a unique strong solution under certain regular conditions on their coefficients. Furthermore, the solutions of this equation depend continuously on the initial value and on the kernels <i>k</i>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_566_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(k_B\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>B</mi> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_566_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(k_Z\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>Z</mi> </msub> </math></EquationSource> </InlineEquation>. We will then show the regularity of solutions for these equations. Finally, we propose a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_566_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-Euler-Maruyama approximation scheme for these equations and demonstrate its convergence at a certain rate in the <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40306_2025_566_Article_IEq6.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>-norm. Some numerical simulations is also presented to support for the theoretical results.</p>

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Well-posedness, Regularity of Solutions and the \(\theta \)-Euler-Maruyama Scheme for Stochastic Volterra Integral Equations with General Singular Kernels and Jumps

  • 期刊论文

  • 发表日期：

    2025年4月24日

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  • Phan Thi Huong,
  • Hoang-Long Ngo,
  • Peter Kloeden

摘要

In this paper, we consider a class of stochastic Volterra integral equations with general singular kernels, driven by a Brownian motion and a pure jump Lévy process. We first show that these equations have a unique strong solution under certain regular conditions on their coefficients. Furthermore, the solutions of this equation depend continuously on the initial value and on the kernels k, \(k_B\) k B , and \(k_Z\) k Z . We will then show the regularity of solutions for these equations. Finally, we propose a \(\theta \) θ -Euler-Maruyama approximation scheme for these equations and demonstrate its convergence at a certain rate in the \(L^2\) L 2 -norm. Some numerical simulations is also presented to support for the theoretical results.