Perturbed Nonlocal Stochastic Functional Differential Equations
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Perturbed Nonlocal Stochastic Functional Differential Equations Qi Zhang1 · Yong Ren2 Received: 5 March 2020 / Accepted: 28 August 2020 © Springer Nature Switzerland AG 2020
Abstract This paper discusses the asymptotic behavior of the solution for a class of perturbed nonlocal stochastic functional differential equations (SFDEs, in short). By comparing it with the solution of the corresponding unperturbed one, we derive the conditions under which their solutions are close. Firstly, the results are established on finite timeintervals. Then, we also show the results hold when the length of time-interval tends to infinity as small perturbations tend to zero. Keywords Nonlocal stochastic functional differential equation · Small perturbation · Closeness Mathematics Subject Classification 60H10 · 60H20 · 34K50
1 Introduction For the practical applications in mechanics, medicine biology, ecology and so on, stochastic functional differential equations (SFDEs, in short) attracted researchers’ more attention. One can see [7,9–11,13] and the references therein. Moreover, nonlocal stochastic differential equations have potential application in finance market, one can see [1,8,12,14] for the details. Especially, Wu and Hu [15] introduced the following nonlocal SFDEs with infinite delay with the form
This work is supported by the National Natural Science Foundation of China (11871076).
B
Yong Ren [email protected]; [email protected] Qi Zhang [email protected]
1
Department of Mathematics, Anhui Normal University, Wuhu 241000, China
2
School of Science, Beijing University of Civil Engineering and Architecture, Beijing 100044, China 0123456789().: V,-vol
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Q. Zhang, Y. Ren
dy(t) = g t, yt , ||yt || p dt + σ t, yt , ||yt || p dB(t), t ≥ 0,
(1)
where yt = yt (θ ) =: {y(t + θ ) : θ ∈ (−∞, 0]}, g and σ are two Borel measurable functions defined on the space R+ × BC((−∞, 0]; Rd ) × R+ . For p ≥ 2, || · || p is a norm in the space L p ((−∞, 0] × ; Rd ) with the form ||yt || p =
1/ p
0
E|y(t + θ )| dη(θ ) p
−∞
,
where η is a probability measure and BC((−∞, 0]; Rd ) is the family of bounded continuous functions from (−∞, 0] to Rd with the norm ||ϕ|| = sup−∞ 0, let L p ([−r , 0]; Rd ) denote the family of Rd -valued, Borel measurable functions ψ(s), −r ≤ s ≤ 0, which is equipped with the following norm ||ψ|| L p =
0 −r
1/ p |ψ(s)| ds p
< ∞.
Let BCF0 ([−r , 0]; Rd ) be the family of continuous bounded Rd -valued stochastic process φ = {φ(s), −r ≤ s ≤ 0} such that φ(s) is F0 -measurable for every s, here, we require that Fs = F0 for −r ≤ s ≤ 0. In this paper, we consider the following nonlocal SFDE with delay with the form
dy(t) = g(t, yt , ||yt ||2 ) dt + σ (t, yt , ||yt ||2 ) dB(t), t ≥ 0, y(t) = φ(t), −r ≤ t ≤ 0,
(2)
where yt = yt (θ ) = {y(t + θ ) : −r ≤ θ ≤ 0} is an L 2 ([−r , 0]; Rd )-valued stochastic process, g : R+ × L 2 ([−r , 0]; Rd ) × R+ → Rd and σ : R+ × L 2 ([−r , 0]; Rd ) × R+ → Rd×n are two Borel measurable functions, and || · ||2 is a norm in the space L 2 ([−r
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