<p>For the Euler scheme of the stochastic linear evolution equations, the discrete stochastic maximal <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_375_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$ L^p $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-regularity estimate is established, and a sharp error estimate in the norm <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_375_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="127" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Vert {\cdot } \Vert _{L^p(\Omega \times (0,T);L^q({\mathcal {O}}))} $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mo>·</mo> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> <msup> <mi>L</mi> <mi>q</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">O</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40072_2025_375_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$ p,q \in [2,\infty ) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is derived via a duality argument.</p>

Stability and convergence of the Euler scheme for stochastic linear evolution equations in Banach spaces

期刊论文
发表日期：

2025年6月29日

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Binjie Li,
Xiaoping Xie

摘要

For the Euler scheme of the stochastic linear evolution equations, the discrete stochastic maximal $ L^p $ $L^{p}$ -regularity estimate is established, and a sharp error estimate in the norm $ \Vert {\cdot } \Vert _{L^p(\Omega \times (0,T);L^q({\mathcal {O}}))} $ ${‖ \cdot ‖}_{L^{p} (Ω \times (0, T); L^{q} (O))}$ , $ p,q \in [2,\infty ) $ $p, q \in [2, \infty)$ , is derived via a duality argument.