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May 2020 Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes
Richard A. Davis, Mikkel Slot Nielsen, Victor Rohde
Bernoulli 26(2): 799-827 (May 2020). DOI: 10.3150/18-BEJ1086

Abstract

In this paper, we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyperbolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs.

Citation

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Richard A. Davis. Mikkel Slot Nielsen. Victor Rohde. "Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes." Bernoulli 26 (2) 799 - 827, May 2020. https://doi.org/10.3150/18-BEJ1086

Information

Received: 1 July 2018; Revised: 1 October 2018; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166548
MathSciNet: MR4058352
Digital Object Identifier: 10.3150/18-BEJ1086

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 2 • May 2020
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