<p>In this paper, we explore strong rate of convergence of Euler-Maruyama scheme for a class of stochastic differential equations with state-dependent Markovian switching driven by multiplicative <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)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Both drift coefficient and the coefficient in front of the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable noise are Lipschitz continuous and uniformly bounded. We demonstrate that the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> convergence rate of the Euler-Maruyama approximation to the true solution is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\Delta |\ln \Delta |)^{1/\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">|</mo> <mo>ln</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>. Numerical example is furnished to illustrate and validate the theoretical findings.</p>

Strong convergence rate of Euler-Maruyama scheme for SDEs with state-dependent Markovian switching driven by multiplicative $\alpha $-stable process

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发表日期：

2026年6月24日

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Liqiong Wang,
Yaozhong Hu,
Qing Zhou

摘要

In this paper, we explore strong rate of convergence of Euler-Maruyama scheme for a class of stochastic differential equations with state-dependent Markovian switching driven by multiplicative $\alpha $ $α$ -stable process, where $\alpha \in (1,2)$ $α \in (1, 2)$ . Both drift coefficient and the coefficient in front of the $\alpha $ $α$ -stable noise are Lipschitz continuous and uniformly bounded. We demonstrate that the $L^{1}$ $L^{1}$ convergence rate of the Euler-Maruyama approximation to the true solution is $(\Delta |\ln \Delta |)^{1/\alpha }$ ${(Δ | ln Δ |)}^{1 / α}$ . Numerical example is furnished to illustrate and validate the theoretical findings.

Strong convergence rate of Euler-Maruyama scheme for SDEs with state-dependent Markovian switching driven by multiplicative \(\alpha \)-stable process

摘要