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<p>The Rosenblatt process is a non-Gaussian self-similar process residing in the second Wiener chaos. It emerges as the limit of correlated random sequences in "non-central limit theorems." It shares the same covariance function as fractional Brownian motion. In this paper, we studied a class of one-dimensional stochastic differential equations driven by the Rosenblatt process with Hurst parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\frac{1}{2}&lt;H&lt;1.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&lt;</mo> <mi>H</mi> <mo>&lt;</mo> <mn>1</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We proved the existence and uniqueness of solution to this kind of equation under the linear growth and Lipschitz conditions. Additionally, using the Gronwall inequality lemma, we provide the continuous dependency of solutions on the initial value. We solve numerically our SDE using Euler Maruyama method. Finally the approximation solution are compared with the exact solution for different sample paths in an example.</p>

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Stochastic differential equation driven by the Rosenblatt process

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月31日

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  • Sakina Benkaddour,
  • Abdeldjebber Kandouci,
  • Omar Kebiri,
  • Tomàs Caraballo

摘要

The Rosenblatt process is a non-Gaussian self-similar process residing in the second Wiener chaos. It emerges as the limit of correlated random sequences in "non-central limit theorems." It shares the same covariance function as fractional Brownian motion. In this paper, we studied a class of one-dimensional stochastic differential equations driven by the Rosenblatt process with Hurst parameter \(\frac{1}{2}<H<1.\) 1 2 < H < 1 . We proved the existence and uniqueness of solution to this kind of equation under the linear growth and Lipschitz conditions. Additionally, using the Gronwall inequality lemma, we provide the continuous dependency of solutions on the initial value. We solve numerically our SDE using Euler Maruyama method. Finally the approximation solution are compared with the exact solution for different sample paths in an example.