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Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes

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Stochastic differential equations with a fractionally filtered delay : A semimartingale model for long-range dependent processes. / Davis, Richard A.; Nielsen, Mikkel Slot; Rohde, Victor Ulrich.

I: Bernoulli, Bind 26, Nr. 2, 2020, s. 799-827.

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avisTidsskriftartikelForskningpeer review

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Davis, Richard A. ; Nielsen, Mikkel Slot ; Rohde, Victor Ulrich. / Stochastic differential equations with a fractionally filtered delay : A semimartingale model for long-range dependent processes. I: Bernoulli. 2020 ; Bind 26, Nr. 2. s. 799-827.

Bibtex

@article{182404b75dc2455ab00cd171ee97f333,
title = "Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes",
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.",
author = "Davis, {Richard A.} and Nielsen, {Mikkel Slot} and Rohde, {Victor Ulrich}",
year = "2020",
doi = "10.3150/18-BEJ1086",
language = "English",
volume = "26",
pages = "799--827",
journal = "Bernoulli",
issn = "1350-7265",
publisher = "International Statistical Institute",
number = "2",

}

RIS

TY - JOUR

T1 - Stochastic differential equations with a fractionally filtered delay

T2 - A semimartingale model for long-range dependent processes

AU - Davis, Richard A.

AU - Nielsen, Mikkel Slot

AU - Rohde, Victor Ulrich

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

U2 - 10.3150/18-BEJ1086

DO - 10.3150/18-BEJ1086

M3 - Journal article

VL - 26

SP - 799

EP - 827

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -