Almost sure Convergence of Solutions of Linear Stochastic Volterra Equations to Nonequilibrium Limits

Appleby, John A. D.; Devin, Siobhán; Reynolds, David W.

doi:10.1216/jiea/1192628616

Cited by 12 publications

(11 citation statements)

References 9 publications

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“…The square integrability of the discrepancy between the solution and the limit is also studied. The results obtained in [3] and [4] are special cases of the ones found here, where the functional G in (1.1) is of the special form [15] considered the necessary and sufficient conditions for asymptotic convergence of solutions of (1.2) to a nontrivial limit and the integrability of these solutions. Before recalling their main result we define the following notation introduced in [15] and adopted in this paper.…”

Section: Convergence To a Non-equilibrium Limitmentioning

confidence: 98%

“…However, the condition (3.6) is required. The condition (3.5) is exactly that required for mean-square convergence in [4], and for almost sure convergence in [3]. In both these papers, the functional G depends only on t, and not on the path of the solution.…”

Section: 2mentioning

confidence: 99%

“…This paper studies the asymptotic convergence of the solution of the stochastic functional differential equation dX(t) = AX(t) + t 0 K(t − s)X(s) ds dt + G(t, X t ) dB(t), t > 0, (1.1a) X(0) = X 0 , (1.1b) to a non-equilibrium limit. The paper develops recent work in Appleby, Devin and Reynolds [3,4], which considers convergence to nonequilibrium limits in linear stochastic Volterra equations. The distinction between the works is that in [3,4], the diffusion coefficient is independent of the solution, and so it is possible to represent the solution explicitly; in this paper, such a representation does not apply.…”

Section: Introductionmentioning

confidence: 99%

“…The paper develops recent work in Appleby, Devin and Reynolds [3,4], which considers convergence to nonequilibrium limits in linear stochastic Volterra equations. The distinction between the works is that in [3,4], the diffusion coefficient is independent of the solution, and so it is possible to represent the solution explicitly; in this paper, such a representation does not apply. However, we can avail of a variation of constants argument, which enables us to prove that the solution is bounded in mean square.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Convergence rates of the solution of a Volterra-type stochastic differential equations to a non-equilibrium limit

Appleby¹

2007

Proceedings of the 8'th Colloquium on the Qualitative Theory of Differential Equations (June 25--28, 2007, Szeged, Hungary) Edi

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This paper concerns the asymptotic behaviour of solutions of functional differential equations with unbounded delay to non-equilibrium limits. The underlying deterministic equation is presumed to be a linear Volterra integro-differential equation whose solution tends to a non-trivial limit. We show when the noise perturbation is bounded by a non-autonomous linear functional with a square integrable noise intensity, solutions tend to a non-equilibrium and non-trivial limit almost surely and in meansquare. Exact almost sure convergence rates to this limit are determined in the case when the decay of the kernel in the drift term is characterised by a class of weight functions.

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Section: Convergence To a Non-equilibrium Limitmentioning

confidence: 98%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Convergence rates of the solution of a Volterra-type stochastic differential equations to a non-equilibrium limit

Appleby¹

2007

Proceedings of the 8'th Colloquium on the Qualitative Theory of Differential Equations (June 25--28, 2007, Szeged, Hungary) Edi

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“…It was shown that under the condition that the kernel does not change sign on 0, ∞ then i the almost sure exponential convergence of the solution to zero, ii the pth mean exponential convergence of the solution to zero, and iii the exponential integrability of the kernel and the exponential square integrability of the noise are equivalent. Two papers by Appleby et al 8,9 considered the convergence of solutions of 1.4a -1.4b to a nonequilibrium limit in the mean square and almost sure senses, respectively. Conditions on the resolvent, kernel, and noise for the convergence of solutions to an explicit limiting random variable were found.…”

Section: K T − S X S Ds F T Dt σ T Db T T > 0 11amentioning

confidence: 99%

Characterisation of Exponential Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

Appleby

Devin

Reynolds

2008

International Journal of Stochastic Analysis

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A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

et al. 2022

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The deflection of Euler–Bernoulli beams under stochastic dynamic loading, exhibiting purely viscous behavior, is characterized by partial differential equations of the fourth order. This paper proposes a computational method to determine the approximate solution to such equations. The functions are approximated using two‐dimensional shifted Legendre polynomials. An operational matrix of integration and an operational matrix of stochastic integration are derived. The operational matrices assist in breaking down the problem under consideration into a set of algebraic equations that may be solved using any known numerical technique that leads to the solution of the stochastic beam equation. The well‐posedness of the problem is studied. The proposed methodology is demonstrated to be practical for addressing the novel stochastic dynamic loading problem by confirming the outcome using a few numerical examples. Thus the effectiveness and applicability of the technique are ensured. The solution quality is explored through diagrams. The accuracy of the method is substantiated by comparing it with the Runge–Kutta method of order 1.5 (R–K 1.5). The absolute error caused by the proposed technique is comparably much less than R–K 1.5. A simulation analysis is carried out with MATLAB, and an algorithm is developed.

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Almost sure Convergence of Solutions of Linear Stochastic Volterra Equations to Nonequilibrium Limits

Cited by 12 publications

References 9 publications

Convergence rates of the solution of a Volterra-type stochastic differential equations to a non-equilibrium limit

Convergence rates of the solution of a Volterra-type stochastic differential equations to a non-equilibrium limit

Characterisation of Exponential Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

A numerical solution for a stochastic beam equation exhibiting purely viscous behavior

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