Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases Article Swipe

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YOU? · · 2024 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2407.07767

In this article we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are $p$-summable along with when their continuous time counterparts, perturbed linear stochastic Volterra integrodifferential equations are $p$-integrable. In the discrete case we find it necessary and sufficient that perturbing functions are $p$-summable in order to ensure paths of the discrete equation are $p$-summable almost surely, while in the continuous case it transpires one can have almost surely $p$-integrable sample paths even with non-integrable perturbation functions. For the continuous equation the main converse is clinched by considering an appropriate discretisation and applying results from the discrete case. We also conduct a thorough study of the asymptotic behaviour of the trajectories of solutions to the continuous equation in the regime of $p$-integrable paths and provide a characterisation of asymptotic convergence to zero in the case of diagonal noise. Additionally we highlight how all proof methods can be applied to obtain even stronger results for stochastic functional differential equations.

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Convolution (computer science) Mathematics Volterra equations Space (punctuation) Mathematical analysis Applied mathematics Volterra integral equation Pure mathematics Nonlinear system Physics Computer science Integral equation Quantum mechanics Machine learning Operating system Artificial neural network

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2407.07767
PDF: https://arxiv.org/pdf/2407.07767
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Related Works: 10
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DOI: https://doi.org/10.48550/arxiv.2407.07767

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Title: Solution space characterisation of perturbed linear discrete and continuous stochastic Volterra convolution equations: the $\ell^p$ and $L^p$ cases

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Type: preprint

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Language: en

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Publication year: 2024

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Publication date: 2024-07-10

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Authors: John A. D. Appleby, Emmet Lawless

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Landing page: https://arxiv.org/abs/2407.07767

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Concepts: Convolution (computer science), Mathematics, Volterra equations, Space (punctuation), Mathematical analysis, Applied mathematics, Volterra integral equation, Pure mathematics, Nonlinear system, Physics, Computer science, Integral equation, Quantum mechanics, Machine learning, Operating system, Artificial neural network

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