<p>In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Hölder continuous in time and bounded <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-Hölder continuous in space with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha ,\beta \in (0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. The strong order of convergence of the randomised EM in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-norm is shown to be <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$1/2+(\alpha \wedge (\beta /2))-\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>-</mo> <mi>ϵ</mi> </mrow> </math></EquationSource> </InlineEquation> for an arbitrary <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\epsilon \in (0,1/2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, higher than the one of standard EM, which is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha \wedge (1/2+\beta /2-\epsilon )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∧</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mi>β</mi> <mo stretchy="false">/</mo> <mn>2</mn> <mo>-</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.</p>

Randomised Euler-Maruyama method for SDEs with Hölder continuous drift coefficient

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2025年11月17日

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Jianhai Bao,
Yue Wu

摘要

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be $\alpha $ $α$ -Hölder continuous in time and bounded $\beta $ $β$ -Hölder continuous in space with $\alpha ,\beta \in (0,1]$ $α, β \in (0, 1]$ . The strong order of convergence of the randomised EM in $L^p$ $L^{p}$ -norm is shown to be $1/2+(\alpha \wedge (\beta /2))-\epsilon $ $1 / 2 + (α \land (β / 2)) - ϵ$ for an arbitrary $\epsilon \in (0,1/2)$ $ϵ \in (0, 1 / 2)$ , higher than the one of standard EM, which is $\alpha \wedge (1/2+\beta /2-\epsilon )$ $α \land (1 / 2 + β / 2 - ϵ)$ . The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.