Projected Euler-Maruyama method for stochastic delay differential equations under a global monotonicity condition

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Abstract

In this paper, we investigate a projected Euler-Maruyama method for stochastic delay differential equations with variable delay under a global monotonicity condition. This condition admits some equations with highly nonlinear drift and diffusion coefficients. We appropriately generalize the idea of C-stability and B-consistency given by Beyn et al. (2016) to the case with delay. Moreover, the method is proved to be convergent with order one-half in a succinct way. Finally, some numerical examples are included to support our theoretical results.

Introduction

Consider d-dimensional nonlinear stochastic delay differential equations (SDDEs) with variable lagdY(t)=f(Y(t),Y(tτ(t)))dt+g(Y(t),Y(tτ(t)))dW(t),t>0,Y(t)=ξ(t)C([τ,0];Rd),t[τ,0].Here, the drift term f:Rd×RdRd and diffusion term g:Rd×RdRd×m are continuous functions. W(t):=(W1(t),,Wm(t))T is an m-dimensional Wiener process defined on given complete probability space (Ω,F,P) with a filtration {Ft}t0 under usual condition. Continuous function τ(t) satisfies τ(t) ≥ τ0 > 0 and τ:=mint0(tτ(t)). And C([τ,0];Rd) denotes the space of continuous functions from [τ,0] to Rd with the norm ψ=supτt0|ψ(t)|. 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] byti=ih,h=TN.And denote by C constants independent of h and it may have different values in different places. For x1,x2,xRd, let ⟨x1, x2⟩ and |x| be the Euclidean inner product and the Euclidean norm on Rd, respectively. If Q is a d × m matrix, we let |Q|=trace(QTQ) be its trace norm. For given delay τ(t) and stepsize h, definemi˜=τ(ti)/h,i=0,1,,so that there exist a unique integer mi and a real number δi ∈ [0, 1) such that mi˜=mi+δi

Convergence theorem

Lemma 3.1

Let Assumptions 1 and 2 hold, and suppose the exact solution of (1.1) satisfies supt[τ,T]Y(t)L4q2(Ω;Rd)<, then we haveE|Y(sτ(s))Y(tiτ(ti))|2Ch,E|Y(sτ(s))Y(ti+1τ(ti+1))|2Ch,for s[ti,ti+1], 0iN1.

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 κ=12. To this end, we first recall a result on the projection function defined in (2.2).

Lemma 4.1

[26]

The projection function x¯ defined by (2.2) satisfies that, for any α ∈ (0, ∞), h ∈ (0, 1] and x1,x2Rd,|x¯1x¯2||x1x2|.

Lemma 4.2

If Assumption 1 is fulfilled with L ∈ (0, ∞), q ∈ (1, ∞) and ζ(12,), then there exists a constant C such that for all random variables y1,z1,y2,z2L2(Ω,Ft,P;Rd),|y¯1z¯1+h(f(y

Numerical experiments

Example 1

We consider the following example [24]dY(t)=[2Y(t)+Y(t1)Y5(t)]dt+Y2(t)dW(t),t0,where Y(t)=cos(t) for 1t0.

Example 2

Next, we consider a more complex SDDE with a nonlinear delay termdY(t)=[2Y(t)+Y(tτ(t))Y5(t)]dt+[Y2(t)+sin(Y(t))sin(Y(tτ(t)))]dW(t),t0,where Y(t)=cos(t) for 1t0 and τ(t)=11+t2.

One can see that Examples 1 and 2 satisfy condition (2.5), Assumption 1 with q=5 and Assumption 4 with p=30. Obviously, p6q4. Hence, we can choose the projection parameter α=12(51)=18. We use M¯=2000

Acknowledgments

The authors are grateful to the anonymous referees and the editors for their valuable comments and suggestions.

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This work was supported by National Natural Science Foundation of China (No. 11771163).

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