Limit theorems for additive functionals of stochastic functional differential equations with infinite delay
type: Item
instance of: scholarly article
title: Limit theorems for additive functionals of stochastic functional differential equations with infinite delay
zbMATH Open document ID: 1481.60073
author: Ya Wang
author: Fuke Wu
author: Chao Zhu
published in: Journal of Differential Equations
publication date: 2021-11-30T00:00:00Z
review text: For a given \(r > 0\), define \({C_r} = \{ \phi \in C(( - \infty ,0];{\mathbb R^d}):\mathop {\sup }\limits_{ - \infty < \theta \le 0} {e^{r\theta }}\left| {\phi (\theta )} \right| < \infty \} \) with norm \({\left\| \phi \right\|_r} = \mathop {\sup }\limits_{ - \infty < \theta \le 0} {e^{r\theta }}\left| {\phi (\theta )} \right|\), where \(C(( - \infty ,0];{\mathbb R^d})\) denotes the family of continuous functions from \(( - \infty ,0]\) to \({\mathbb R^d}\). Choosing \({C_r}\) as the phase space, the paper considers the stochastic functional differential equations (SFDEs) with infinite delay \(dX(t) = b({X_t})dt + \sigma ({X_t})dW(t),{X_0} = \xi \in {C_r}\), where \(b:{C_r} \to {\mathbb R^d}\) and \(\sigma :{C_r} \to {\mathbb R^{d \times m}}\) are continuous functionals, \(W(t)\) is an \(m\)-dimensional Wiener process on a compete filtered probability space and \({X_t}(\theta ):\theta \mapsto X(t + \theta ) \in {\mathbb R^d},\theta \in ( - \infty ,0]\) denotes the segment process. The paper aims to investigate the large-time behavior of the functional \({t^{ - 1}}\int_0^t {f({X_s}(\xi )ds} \) as \(t \to \infty \). It establishes the strong law of large numbers and the central limit theorem for SFDEs with non-Lipschitz coefficients and infinite delay. The paper also establishes the law of iterated logarithm for SFDEs with infinite delays under some dissipative condition.
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 7436217
zbMATH Keywords: stochastic functional differential equations with infinite delay
zbMATH Keywords: non-Lipschitz conditions
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: central limit theorem
zbMATH Keywords: law of the iterated logarithm
reviewed by: Oleg K. Zakusilo
MaRDI profile type: Publication
full work available at URL: j.jde.2021.11.011
OpenAlex ID: W3213359040
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DOI: 10.1016/J.JDE.2021.11.011
description: scientific article; zbMATH DE number 7436217