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<p>This paper investigates both qualitative and quantitative aspects of a class of singular stochastic Volterra integral equations with jumps (SSVIEJ). On the qualitative aspect, we establish fundamental results on the existence and uniqueness of solutions, together with their continuous dependence on initial data and on the singularity degrees of the kernel functions. We then examine the temporal Hölder continuity of solutions and derive the optimal asymptotic separation rate between distinct solutions. On the quantitative aspect, we develop and analyze several discretization schemes, including the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-Euler–Maruyama (EM) method and its accelerated variant, the fast <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\theta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>-EM scheme. We investigate their strong convergence properties, convergence rates, and computational complexities. Furthermore, in the case without jumps, we analyze the strong convergence under the supremum norm by employing the Garsia–Rodemich–Rumsey inequality. The theoretical results are supported by numerical simulations, which demonstrate the effectiveness and accuracy of the proposed methods.</p>

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Asymptotic separation between solutions and fast \(\theta \)-Euler-Maruyama scheme for singular stochastic Volterra integral equations with jumps

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Phan Thi Huong,
  • Hoang-Long Ngo

摘要

This paper investigates both qualitative and quantitative aspects of a class of singular stochastic Volterra integral equations with jumps (SSVIEJ). On the qualitative aspect, we establish fundamental results on the existence and uniqueness of solutions, together with their continuous dependence on initial data and on the singularity degrees of the kernel functions. We then examine the temporal Hölder continuity of solutions and derive the optimal asymptotic separation rate between distinct solutions. On the quantitative aspect, we develop and analyze several discretization schemes, including the \(\theta \) θ -Euler–Maruyama (EM) method and its accelerated variant, the fast \(\theta \) θ -EM scheme. We investigate their strong convergence properties, convergence rates, and computational complexities. Furthermore, in the case without jumps, we analyze the strong convergence under the supremum norm by employing the Garsia–Rodemich–Rumsey inequality. The theoretical results are supported by numerical simulations, which demonstrate the effectiveness and accuracy of the proposed methods.