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

John A. D. Appleby, Emmet Lawless

TL;DR

This work characterises when perturbed linear stochastic Volterra equations possess almost surely $p$-integrable trajectories in continuous time and $p$-summable paths in discrete time. By linking the Volterra dynamics to an associated OU process through resolvent techniques, the authors derive sharp, checkable conditions on the forcing term $f$ and the noise coefficient $\sigma$ that govern $p$-integrability, expressed as time-average integral conditions or discrete-time summability depending on $p$. The discrete and continuous theories yield equivalences between pathwise $L^p$-integrability and the $L^p$-properties of $(f,\sigma)$, with a detailed treatment of Gaussian noise and a parallel theory for stochastic functional differential equations that confirms memory does not alter the $p$-integrability character. The paper also provides asymptotic convergence results in the diagonal-noise case and supports the theory with explicit examples, showing the conditions are both necessary and sufficient and broadly applicable to memory-rich stochastic models.

Abstract

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.

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

TL;DR

This work characterises when perturbed linear stochastic Volterra equations possess almost surely -integrable trajectories in continuous time and -summable paths in discrete time. By linking the Volterra dynamics to an associated OU process through resolvent techniques, the authors derive sharp, checkable conditions on the forcing term and the noise coefficient that govern -integrability, expressed as time-average integral conditions or discrete-time summability depending on . The discrete and continuous theories yield equivalences between pathwise -integrability and the -properties of , with a detailed treatment of Gaussian noise and a parallel theory for stochastic functional differential equations that confirms memory does not alter the -integrability character. The paper also provides asymptotic convergence results in the diagonal-noise case and supports the theory with explicit examples, showing the conditions are both necessary and sufficient and broadly applicable to memory-rich stochastic models.

Abstract

In this article we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are -summable along with when their continuous time counterparts, perturbed linear stochastic Volterra integrodifferential equations are -integrable. In the discrete case we find it necessary and sufficient that perturbing functions are -summable in order to ensure paths of the discrete equation are -summable almost surely, while in the continuous case it transpires one can have almost surely -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 -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.
Paper Structure (14 sections, 19 theorems, 152 equations, 1 figure)

This paper contains 14 sections, 19 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Discrete results
  3. Notation and mathematical preliminaries
  4. Results for general noise
  5. Gaussian noise
  6. Continuous Results
  7. Notation
  8. Main Results
  9. Almost sure asymptotic behaviour
  10. Stochastic Functional Differential Equations
  11. Examples
  12. Conclusion
  13. End Notes
  14. Appendix

Key Result

Proposition 1

Let $\xi$ be a real valued random variable whose distribution has either an absolutely continuous part or at least two isolated atoms (with non-zero probability), then $\xi \in \mathbb{D}$.

Figures (1)

  • Figure 1: Graph of $g(t)$ with $\beta=0.32$

Theorems & Definitions (40)

  • Definition 1
  • Remark 1
  • Proposition 1
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof : Proof of Lemma \ref{['lem. summability converse']}
  • proof : Proof of Theorem \ref{['thm. lp characterisation of Volterra difference equation']}
  • Theorem 3
  • Lemma 2
  • ...and 30 more