<p>The contraction mapping principle (CMP, i.e. the fixed point technique of Banach–Caccioppoli) is used to show existence and uniqueness of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-solutions of Itô-type stochastic neutral integro-differential equations with infinite memory, driven by a standard Wiener process. For this purpose, we study several properties of an associated integral-type map <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {H}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">H</mi> </math></EquationSource> </InlineEquation>, which is a contraction on the strong Banach space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathbb {S}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">S</mi> </math></EquationSource> </InlineEquation> of adapted, stochastic processes with finite supremum moments and appropriate contraction constant. As a side-product, the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-error of appropriate successive iterations is estimated and simulation-results of a numerical example are given. Properties of the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-solutions such as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>- and a.s. Hölder-continuity, and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-boundedness are investigated.</p>

Existence, uniqueness and Hölder-continuity of $L^p\!$-solutions of stochastic neutral integro-differential equations with infinite memory

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发表日期：

2025年7月14日

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Saeed Althubiti,
Henri Schurz

摘要

The contraction mapping principle (CMP, i.e. the fixed point technique of Banach–Caccioppoli) is used to show existence and uniqueness of $L^p$ $L^{p}$ -solutions of Itô-type stochastic neutral integro-differential equations with infinite memory, driven by a standard Wiener process. For this purpose, we study several properties of an associated integral-type map ${\mathbb {H}}$ $H$ , which is a contraction on the strong Banach space ${\mathbb {S}}$ $S$ of adapted, stochastic processes with finite supremum moments and appropriate contraction constant. As a side-product, the $L^p$ $L^{p}$ -error of appropriate successive iterations is estimated and simulation-results of a numerical example are given. Properties of the $L^p$ $L^{p}$ -solutions such as $L^p$ $L^{p}$ - and a.s. Hölder-continuity, and $L^p$ $L^{p}$ -boundedness are investigated.

Existence, uniqueness and Hölder-continuity of \(L^p\!\)-solutions of stochastic neutral integro-differential equations with infinite memory

摘要