Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition☆
Introduction
Consider d-dimensional nonlinear stochastic delay differential equations (SDDEs) with variable lagHere, the drift term and diffusion term are continuous functions. is an m-dimensional Wiener process defined on given complete probability space with a filtration under usual condition. Continuous function τ(t) satisfies τ(t) ≥ τ0 > 0 and . And denotes the space of continuous functions from to with the norm . SDDEs can be considered as an extension of stochastic ordinary differential equations (SODEs), and they play a key role in many phenomena in physics [1], [2], [3]. The well-posedness of the equation is well established under the global Lipschitz condition and the linear growth condition (see [4], [5]). In 2002, Mao [6] replaced the linear growth condition with weaker Khasminskii-type condition and discussed the well-posedness of SDDEs. Generally, SDDEs cannot be solved analytically, so that numerical simulation is particularly necessary.
In the past several decades, a number of numerical methods were investigated for SDDEs with one-sided Lipschitz drift and linearly growing diffusion coefficients (see [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and references therein). Limited work has been done in SDDEs whose diffusion coefficients do not satisfy the linear growth condition, and this issue received attention only recently. For example, Huang [20] discussed the mean square stability and dissipativity of two classes of theta methods for SDDEs. Yan and Xiao [21] investigated the split-step theta method for neutral stochastic delay differential equations with polynomial growth delay terms. Dareiotis et al. [22] generalized the tamed Euler-Maruyama (EM) approach to a kind of nonlinear SDDEs driven by Lévy noise. In 2018, Guo et al. [23] considered the truncated EM method for nonlinear SDDEs under the generalized Khasminskii-type condition. Zhang et al. [24] studied partially truncated EM method for a class of highly nonlinear SDDEs. All their convergence analyses were under the framework given by Higham et al. [25], where moment-bounded proof and continuous time extension of the numerical solution need to be taken into account.
In this paper, we consider projected EM method for SDDEs with variable lag. This method is an extension of the one for SODEs in [26]. We generalize the notions of C-stability and B-consistency for SODEs introduced by Beyn et al. [26], [27], and then prove that the projected EM method is strongly convergent of order one-half for SDDEs with some superlinear growth conditions. We mention that we do not need the moment estimates of the numerical solution in the proof.
The rest of this paper is organized as follows. Some assumptions and projected EM method are introduced in Section 2. Section 3 gives a general convergence theorem based on the premise of C-stability and B-consistency. In Section 4, C-stability and B-consistency of the projected EM method are analyzed in detail. In Section 5, some numerical experiments are displayed to support our theoretical analyses.
Section snippets
Preliminaries
Throughout this paper, we define a uniform mesh on [0, T] byAnd denote by C constants independent of h and it may have different values in different places. For let ⟨x1, x2⟩ and |x| be the Euclidean inner product and the Euclidean norm on respectively. If Q is a d × m matrix, we let be its trace norm. For given delay τ(t) and stepsize h, defineso that there exist a unique integer mi and a real number δi ∈ [0, 1) such that
Convergence theorem
Lemma 3.1 Let Assumptions 1 and 2 hold, and suppose the exact solution of (1.1) satisfies then we havefor . Proof The main idea of the proof is similar to that of Lemma 4.4 in Section 4. So it will not be reproduced here. □
Now we prove a general result that C-stability plus B-consistency implies mean square convergence, which is an extension of the result on SODEs given in [26]. Theorem 3.1 If the one-step method
Stability and consistency analysis
In what follows we will verify that the projected EM method is stochastically C-stable and B-consistent of order . To this end, we first recall a result on the projection function defined in (2.2). Lemma 4.1 The projection function defined by (2.2) satisfies that, for any α ∈ (0, ∞), h ∈ (0, 1] and Lemma 4.2 If Assumption 1 is fulfilled with L ∈ (0, ∞), q ∈ (1, ∞) and then there exists a constant C such that for all random variables [26]
Numerical experiments
Example 1 We consider the following example [24]where for . Example 2 Next, we consider a more complex SDDE with a nonlinear delay termwhere for and .
One can see that Examples 1 and 2 satisfy condition (2.5), Assumption 1 with and Assumption 4 with . Obviously, . Hence, we can choose the projection parameter . We use
Acknowledgments
The authors are grateful to the anonymous referees and the editors for their valuable comments and suggestions.
References (29)
- et al.
Feedback and delays in neurological diseases: a modeling study using gynamical systems
Bull. Math. Biol.
(1993) - et al.
Convergence and stability of the semi-implicit Euler method for a linear stochastic differential delay equation
J. Comput. Appl. Math.
(2004) Exponential stability of equidistant Euler-Maruyama approximations of stochastic differential delay equations
J. Comput. Appl. Math.
(2007)- et al.
θ-Maruyama methods for nonlinear stochastic differential delay equations
Appl. Numer. Math.
(2015) - et al.
Numerical analysis of explicit one-step methods for stochastic delay differential equations
LMS J. Comput. Math.
(2000) - et al.
On mean-square stability of two-step Maruyama methods for nonlinear neutral stochastic delay differential equations
Appl. Math. Comput.
(2015) - et al.
Theta schemes for SDDEs with non-globally lipschitz continuous coefficients
J. Comput. Appl. Math.
(2015) - et al.
Strong predictor-corrector approximation for stochastic delay differential equations
J. Comput. Math.
(2015) - et al.
Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations
Appl. Numer. Math.
(2011) - et al.
Split-step θ-method for stochastic delay differential equations
Appl. Numer. Math.
(2014)