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<p>In this paper, we investigate the strong and weak approximations of Euler-Maruyama scheme for a class of stochastic differential equations with state-dependent Markovian switching driven by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable process, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\alpha \in (1,2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The strong convergence order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(1/\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> is obtained through direct calculations with state-dependent Markovian switching under Lipschitz continuous drift, and the weak convergence order 1 is got by the technique of Kolmogorov equation with non-state-dependent Markovian switching, where drift coefficient satisfies Lipschitz continuity and other regularity conditions. Numerical example is provided for demonstration.</p>

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Strong and weak convergence rates of Euler-Maruyama scheme for stochastic differential equations with state-dependent Markovian switching driven by \(\alpha \)-stable process

  • 期刊论文

  • 发表日期：

    2025年5月6日

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  • Liqiong Wang,
  • Qing Zhou

摘要

In this paper, we investigate the strong and weak approximations of Euler-Maruyama scheme for a class of stochastic differential equations with state-dependent Markovian switching driven by \(\alpha \) α -stable process, where \(\alpha \in (1,2)\) α ∈ ( 1 , 2 ) . The strong convergence order \(1/\alpha \) 1 / α is obtained through direct calculations with state-dependent Markovian switching under Lipschitz continuous drift, and the weak convergence order 1 is got by the technique of Kolmogorov equation with non-state-dependent Markovian switching, where drift coefficient satisfies Lipschitz continuity and other regularity conditions. Numerical example is provided for demonstration.