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Growth and fluctuation in perturbed nonlinear Volterra equations

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  • Appleby, John A.D.
  • Patterson, Denis D.

Abstract

We develop precise bounds on the growth rates and fluctuation sizes of unbounded solutions of deterministic and stochastic nonlinear Volterra equations perturbed by external forces. The equation is sublinear for large values of the state, in the sense that the state–dependence is negligible relative to linear functions. If an appropriate functional of the forcing term has a limit L at infinity, the solution of the differential equation behaves asymptotically like the underlying unforced equation when L=0, like the forcing term when L=+∞, and inherits properties of both the forcing term and unperturbed or fundamental solution for values of L∈(0,∞). Our approach carries over in a natural way to stochastic equations with additive noise and we treat the illustrative cases of Brownian and Lévy noise.

Suggested Citation

  • Appleby, John A.D. & Patterson, Denis D., 2021. "Growth and fluctuation in perturbed nonlinear Volterra equations," Applied Mathematics and Computation, Elsevier, vol. 396(C).
    • Handle: RePEc:eee:apmaco:v:396:y:2021:i:c:s0096300320308912
    DOI: 10.1016/j.amc.2020.125938
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    References listed on IDEAS

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    1. Appleby, John A. D. & Reynolds, David W., 2003. "Non-exponential stability of scalar stochastic Volterra equations," Statistics & Probability Letters, Elsevier, vol. 62(4), pages 335-343, May.
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    3. Benhabib, Jess & Rustichini, Aldo, 1991. "Vintage capital, investment, and growth," Journal of Economic Theory, Elsevier, vol. 55(2), pages 323-339, December.
    4. Marquardt, Tina & Stelzer, Robert, 2007. "Multivariate CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 96-120, January.
    5. Brockwell, Peter J. & Lindner, Alexander, 2009. "Existence and uniqueness of stationary Lévy-driven CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2660-2681, August.
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