MS-stability of the Euler–Maruyama method for stochastic differential delay equations

doi:10.1016/j.amc.2003.10.015

Applied Mathematics and Computation

Volume 159, Issue 1, 25 November 2004, Pages 127-135

https://doi.org/10.1016/j.amc.2003.10.015 Get rights and content

Abstract

The aim of this paper is to investigate the stability of the Euler–Maruyama method for the stochastic differential equations with time delay. The definition of MS-stability of numerical methods is established. The condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise, and the MS-stability of the numerical scheme is proved.

Introduction

Many of the processes, both natural and man-made, involve time delays. That is, the future state of the system is dependent on some of the past history. Indeed, the stochastic differential delay equations (SDDEs) as the stochastic models appear frequently in applied research, and the related study has received considerable attention. For example, the theory of existence and uniqueness of the solution can be found in [9], [13], and the stability of SDDEs has studied by Kolmanovskii and Myshkis [5], Liu and Xia [8], Mao [9], [10]. Moreover, some new results on the exponential stability were established for the SDDEs in [11].

The general form of the SDDEs is $d X(t)=f(t,X(t),X(t−τ)) d t+g(t,X(t),X(t−τ)) d W(t), t⩾0, X(t)=ψ(t), t∈[−τ,0],$ where τ is a positive fixed delay, W(t) is a d-dimensional standard Wiener process and ψ(t) is a $C([−τ,0]; R^{m})$ -valued initial segment, and the functions $f: R^{+} × R^{m} × R^{m} → R^{m}$ , $g: R^{+} × R^{m} × R^{m} → R^{m×d}$ .

Explicit solutions can hardly be obtained for the SDDEs. Thus, it is necessary to develop appropriate numerical methods and to study the properties of these approximate schemes. The convergence and stability properties of the numerical methods for the stochastic ordinary differential equations (SODEs) have been studied by many authors, for instance, Burrage [2], Higham [3], Kloeden and Platen [4], Milstein [12], Saito and Mitsui [14], [15], Tian and Burrage [16]. But the study of numerical scheme for SDDEs is less by far. Concerning the convergence properties, we refer to [1], [6], [7]. However there is no result for the stability of numerical schemes for the SDDEs.

In this paper, we focus on the stability of the Euler–Maruyama numerical solutions for a scalar test equation of the form $d X(t)=[aX(t)+bX(t−τ)] d t+cX(t−τ) d W(t), t⩾0, X(t)=ψ(t), t∈[−τ,0].$ where $a,b,c∈ R$ . The main aim of our paper is to show that the Euler–Maruyama method applied to Eq. (2) is mean square stable under the condition which guarantees the stability of the analytical solution.

In Section 2 we will introduce some necessary notations and assumptions. The condition of the mean square stability of Eq. (2) will be established. In Section 3, the Euler–Maruyama method will be used to produce the numerical solutions. Furthermore, our main result will be shown and proved in this section. We will provide some numerical examples in Section 4.

Section snippets

Mean square stability of analytical solution

Let $(Ω, F,P)$ be a filtered probability space with a filtration $(F_{t})_{t⩾0}$ , which satisfies the usual condition. Let W(t), t⩾0 in Eqs. , be $F_{t}$ -adapted and independent of $F_{0}$ . Moreover, |·| is the Euclidean norm in $R^{m}$ and the matrix norm of matrix g is defined by ∥g∥=sup{|gx|:|x=1|}. We assume the initial segment ψ(t), t∈[−τ,0] to be $F_{0}$ -measurable and right continuous, E∥ψ∥²<∞, where ∥ψ∥=sup_{−τ⩽t⩽0}|ψ(t)|. Further, we let f(t,0,0)=g(t,0,0)≡0. To ensure the existence and uniqueness of the solution, we

Numerical stability analysis

Apply the Euler–Maruyama method to Eq. (2), then we have $X_{n+1} = X_{n} +(a X_{n} +b X_{n−m})h+c X_{n−m} Δ W_{n},$ where h is the stepsize which satisfies τ=mh for an integer number m, and t_n=nh, $X_{n} = X (t_{n})$ , when t_n⩽0, we have $X_{n} =ψ(t_{n})$ . Moreover, the increments ΔW_n:=W(t_n+1)−W(t_n) are independent N(0,h)-distributed Gaussian random variables. We assume $X_{n}$ to be $F_{t_{n}}$ -measurable at the mesh-points t_n.

Definition 3

Under the condition (7), a numerical method applied to Eq. (2) is said to be MS-stable, if there exists a h₀(a,b,c)>0, such that

Numerical examples

We use the equation $d X(t)=[aX(t)+bX(t−1)] d t+cX(t−1) d W(t)$ as a test equation, and take ψ(t)=t+1 for t∈[−τ,0] as an initial function, which is used to compute the solution on t∈[0,1], that is $X(t)= e^{∫_{0}^{t}ads} ψ(0)+∫_{0}^{t} bψ(s−1) e^{−∫_{0}^{s}adr} d s+∫_{0}^{t} cψ(s−1) e^{−∫_{0}^{s}adr} d W(s) = e^{at} 1+ b a^{2} − b a^{2} − b a t+c e^{at} ∫_{0}^{t} s e^{−as} d W(s).$ We obtain the explicit solution on the second interval [1,2] by using the above solution as the new initial function. In fact, on every interval the computing method is the same as the method for SODEs. We compute

References (16)

E. Buckwar
Introduction to the numerical analysis of stochastic delay differential equations
J. Comput. Appl. Math
(2000)
K. Burrage et al.
High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations
Appl. Numer. Math
(1996)
U. Küchler et al.
Strong discrete time approximation of stochastic differential equations with time delay
Math. Comput. Simulat
(2000)
U. Küchler et al.
Weak discrete time approximation of stochastic differential equations with time delay
Math. Comput. Simulat
(2002)
K. Liu et al.
On the exponential stability in mean square of neutral stochastic functional differential equations
Syst. Control Lett
(1999)
X. Mao
Razumikhin-type theorems on exponential stability of stochastic functional differential equations
Stoch. Process. Appl
(1996)
D.J. Higham
Mean-square and asymptotic stability of the stochastic theta method
SIAM J. Numer. Anal
(2000)
P.E. Kloeden et al.
Numerical Solution of Stochastic Differential Equations
(1992)

There are more references available in the full text version of this article.

Cited by (57)

A delayed deterministic and stochastic SI<inf>R</inf>I<inf>C</inf>V model: Hopf bifurcation and stochastic analysis
2024, Mathematics and Computers in Simulation
In this paper, we present a delayed deterministic and stochastic $S I_{R} I_{C} V$ models to investigate the effects of the white noise intensities and the waning immunity of vaccinated individuals in the evolution of the disease. For the deterministic $S I_{R} I_{C} V$ model, the basic reproduction number $R_{0}$ and the equilibrium points are calculated. The local stability of equilibrium points is analyzed. Particularly, when $R_{0} < 1$ the disease-free equilibrium is locally stable for any positive value of $τ$ . Furthermore, when $R_{0} > 1$ , the local stability and sufficient conditions to ensure the occurrence of Hopf bifurcation for the endemic equilibrium point are established by considering the time delay $τ$ as a bifurcation parameter. For the stochastic $S I_{R} I_{C} V$ model, the conditions of the extinction and persistence of the disease are given by using the stochastic basic reproduction numbers $R_{0}^{s}$ and $R_{0}^{s *}$ . Numerical simulations are presented to enhance our analytical results and contrast the deterministic and stochastic models.
Asymptotic mean square stability of Predictor-Corrector methods for stochastic delay ordinary and partial differential equations
2023, Communications in Nonlinear Science and Numerical Simulation
In this paper, we focus on the asymptotic mean square stability of stochastic delay ordinary and partial differential equations. By virtue of root locus technique, the sufficient and necessary conditions of asymptotic mean square stability for Predictor-Corrector methods are obtained, which are suitable for solving above two types of problems. Furthermore, by regulating the value of parameter $θ$ in drift term of the schemes, we could investigate a series of schemes to explore different size of stable regions, this amounts to provide a road to obtain the optimal stable regions. Several theorems prove that the Predictor-Corrector schemes could inherit the asymptotic mean square stability for above two original problems. Numerical experiments verify the stability theorems, precision and the parallelism.
Simulation of two-point reactor kinetics model of reflected reactors with Newtonian reactivity feedback
2022, Annals of Nuclear Energy
In this study, a non-analog dynamic Monte Carlo method, named Point Kinetics Monte Carlo (PKMC), is developed to solve the one- and two-point reactor kinetic equations for a reflected reactor with six group of delayed neutron precursors with Newtonian reactivity feedback. Dynamic weight window technique is implemented to control neutron and precursor population during the simulation. Moreover, the Stochastic differential equations for Two-Point Reactor Kinetics Model (STPRKM) is developed to determine neutron and precursor population as a function of time. Since, the random fluctuations in the population dynamics carry very often as much information as the mean value, during the derivation of STPRKM, physical nature of the model problem and inherent probabilistic mechanisms are not altered. Using the developed methods, time dependent transient problems of reflected reactors are studied and mean responses and standard deviations determined by these methods are compared with each other and available deterministic results.
The convergence of a numerical scheme for additive fractional stochastic delay equations with [Formula presented]
2022, Mathematics and Computers in Simulation
In this paper, we investigate the strong convergence of the exponential Euler method to stochastic delay differential equations with fractional Brownian motion (FSDDEs) of Hurst parameter $H \in (\frac{1}{2}, 1)$ . We establish the strong convergence rate $H$ of the method for FSDDEs to the exact solution. Also we justify our theoretical results with some numerical examples of these equations alongside insignificant step size.
Numerical treatment of stochastic delay differential equations: A global error bound
2019, Journal of Computational and Applied Mathematics
Citation Excerpt :
The state-of-the-art of studying the numerical stability for stochastic differential equations (SDEs) could be found in [13,14]. In the case of SDDE, the mean-square stability of Euler–Maruyama and Milstein methods have been respectively developed in [15,16]. The rest of paper proceeds as follows.
Motivated by the error analysis of numerical solution for stochastic delay differential equations, a new result indicating the dependency of convergence rate of the scheme on two separate quantities $p$ and $q$ , where the former is the global order of underlying scheme and the latter is that of interpolation approximating the history of system, is presented. As an objective instance, the stochastic Runge–Kutta (SRK) schemes of global order 0.5, 1 and 1.5, in the strong convergence sense are utilized to formulate three one-step schemes to approximate the solution of the problem. Moreover, the mean-square stability of these schemes will be asymptotically developed. Finally, the theoretical results will be confirmed by some test problems.
Split-step theta method for stochastic delay integro-differential equations with mean square exponential stability
2019, Applied Mathematics and Computation
In this paper, we propose the split-step theta method for stochastic delay integro-differential equations by the Lagrange interpolation technique and investigate the mean square exponential stability of the proposed scheme. It is shown that the split-step theta method can inherit the mean square exponential stability of the continuous model under the linear growth condition and the proposed stability condition by the delayed differential and difference inequalities established in the paper. A numerical example is given at the end of the paper to illustrate the method and conclusion of the paper. In addition, the convergence of the split-step theta method is proved in the Appendix.

View all citing articles on Scopus

^☆: This work is supported by the NSF of PR China (no. 10271036).

View full text

MS-stability of the Euler–Maruyama method for stochastic differential delay equations☆

Abstract

Introduction

Section snippets

Mean square stability of analytical solution

Numerical stability analysis

Numerical examples

J. Comput. Appl. Math

Appl. Numer. Math

Math. Comput. Simulat

Math. Comput. Simulat

Syst. Control Lett

Stoch. Process. Appl

Mean-square and asymptotic stability of the stochastic theta method

SIAM J. Numer. Anal

Numerical Solution of Stochastic Differential Equations