Numerical Approximation of Stochastic Theta Method for Random Periodic Solution of Stochastic Differential Equations
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Acta Mathemacae Applicatae Sinica, English Series The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
Numerical Approximation of Stochastic Theta Method for Random Periodic Solution of Stochastic Differential Equations Rong WEI1 , Chuan-zhong CHEN2,† 1 School
of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China (E-mail: [email protected]) 2 School
of Science, Hainan University, Haikou 570228, China (E-mail: [email protected])
Abstract
In this paper, we make use of stochastic theta method to study the existence of the numerical
approximation of random periodic solution. We prove that the error between the exact random periodic solution and the approximated one is at the 14 order time step in mean sense when the initial time tends to ∞. Keywords
Stochastic theta method; random periodic solution; numerical approximation
2000 MR Subject Classification
1
34F05; 34K13
Introduction
In this paper, we consider the following stochastic differential equation dXtt0 = [AXtt0 + f (t, Xtt0 )]dt + g(t, Xtt0 )dBt ,
t ≥ t0 .
(1.1)
where Xtt00 = ξ is F t0 measurable, f : R × Rm → Rm , g : R × Rm → Rm×d , A is a negativedefinite m × m symmetric matrix. Throughout this paper, unless otherwise specified, we let (Ω, F, F t , P) be a complete probability space with a filtration F t satisfying the usual conditions (i.e. F t := σ{Bu − Bv : s ≤ v ≤ u ≤ t} is increasing and right continuous while F0 contains all P-null sets). Let B(t) = (Bt1 , · · · , Btm )T be an m-dimensional Brownian motion defined on the probability space. According to [1], the solution of (1.1) is given by ∫ t ∫ t t0 A(t−t0 )ξ At −As t0 At Xt (ξ) = e +e e f (s, Xs )ds + e e−As g(s, Xst0 )dBs . (1.2) t0
t0
Note that u(t, t0 ) given by u(t, t0 )(ξ) = Xtt0 (ξ) satisfies the semi-flow property and the periodic property, respectively, i.e, u(t, r, ω) = u(t, s, ω) ◦ u(s, r, ω) and
u(t + τ, s + τ, w) = u(t, s, θτ ω),
where θ(ω)(s) = B(ω, t + s) − B(ω, t), t, s ∈ R. Let Xt−kτ (ξ, ω) be the solution of (1.1) with initial time −kτ . Then there exists a limit Xt∗ (ξ, ω) for Xt−kτ (ξ, ω) in L2 (Ω) when k tends to ∞, and Xt∗ is the random periodic solution of (1.1)(cf. [1]), which has the following form ∫ t ∫ t ∗ A(t−s) ∗ Xt = e f (s, Xs )ds + eA(t−s) g(s, Xs∗ )dWs . (1.3) −∞
−∞
Manuscript received Match 5, 2019. Accepted on January 17, 2020. This paper is supported by the National Natural Science Foundation of China (No.11871184, 11701127) and by the Natural Science Foundation of Hainan Province(Grant No.117096) † Corresponding author.
R. WEI, C.Z. CHEN
690
In this paper, we denote Tr (A) as the trace of matrix A, N+ is positive natural number, 1 ∥ · ∥p := (E| · |p) p and give the following assumptions. Assumption (I). The eigenvalues {λj , j = 1, 2, · · · , m} of the symmetric matrix A satisfy λm ≤ · · · ≤ λ2 ≤ λ1 < 0. Assumption (II). The coefficients of (1.1) satisfy the periodic property and Lipschitz assumption, i.e, there exists a constant τ > 0 such that for any x ∈ Rm , t ∈ R, f (t + τ, x) = f (t
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