Advances in the Truncated Euler–Maruyama Method for Stochastic Diﬀerential Delay Equations

. Guo et al. [GMY17] are the ﬁrst to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic diﬀerential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [GMY17] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

1. Introduction. Stochastic differential delay equations (SDDEs) have been used in many branches of science and industry (see, e.g., [Arn,CLM01,DZ92,Kha,LL]). The classical theory on the existence and uniqueness of the solution to the SDDE requires the coefficients of the SDDE satisfy the local Lipschitz condition and the linear growth condition (see, e.g., [KM,M97,M02,Moh]). The numerical solutions under the linear growth condition plus the local Lipschitz condition have been discussed intensively by many authors (see, e.g., [BB,BB05,CKR06,DFLM,HM05,KloP,KP,MS,Mil,Schurz,WM08]).
for those x, y,x,ȳ ∈ R n with |x| ∨ |y| ∨ |x| ∨ |ȳ| ≤ R. As we are concerned with highly nonlinear SDDEs, we will not impose the linear growth condition. Instead, we need the Khasminskii-type condition.
Assumption 2.1. There are constants K 1 > 0 andp > 2 such that for all (x, y) ∈ R n × R n .
From now on, we will let the step size ∆ be a fraction of τ . That is, we will use ∆ = τ /M for some positive integer M . When we use the terms of a sufficiently small ∆, we mean that we choose M sufficiently large.
Define t k = k∆ for k = −M, −(M − 1), · · · , 0, 1, 2, · · · . The discrete-time truncated EM solutions are defined by setting X ∆ (t k ) = ξ(t k ) for k = −M, −(M − 1), · · · , 0 and then forming . As in [GMY17], it is more convenient to work on the continuous-time approximations. Recall that there are two continuous-time versions. One is the continuous-time step processx where I [k∆,(k+1)∆) (t) is the indicator function of [k∆, (k + 1)∆) (please recall the notation defined in the beginning of this section). The other one is the continuoustime continuous process We see that x ∆ (t) is an Itô process on t ≥ 0 with its Itô differential (2.11) It is useful to know that X ∆ (t k ) =x ∆ (t k ) = x ∆ (t k ) for every k ≥ −M , namely three of them coincide at t k . [GMY17]. We recall one more notation used in [GMY17]. Let U denote the family of continuous functions U : R n × R n → R + such that for each b > 0, there is a positive constant κ b for which

Review of the main result in
Let us state the assumptions imposed in [GMY17] for the strong convergence rate.

Assumption 2.2.
There is a pair of constants K 2 > 0 and γ ∈ (0, 1] such that the initial data ξ satisfies Assumption 2.3. Assume that there are positive constants α and K 3 and a func- IN TRUNCATED EULER-MARUYAMA METHOD FOR SDDES 5 for all x, y,x,ȳ ∈ R n .
Assumption 2.4. Assume that there is a pair of positive constants r and K 4 such that The following theorem is one of the main results in [GMY17].
Hence, for α = 2.375, We have hence verified Assumption 2.3 with α = 2.375, K 3 = 2.25 and U being defined above. It is also straightforward to show that Assumption 2.4 is satisfied with r = 4 (and some K 4 which is not important).
To apply Theorem 2.5, we still need to design functions µ and h satisfying (2.4) and (2.5), respectively. Note that We can hence have µ(u) = 10u 3 and its inverse function µ −1 (u) = (u/10) 1/3 for u ≥ 0. We also define h(∆) = ∆ −1/5 for ∆ ∈ (0, ∆ * ], where ∆ * = 10 −5 (so h(∆ * ) = 10 = µ(1) as required). Then condition (2.14) becomes ∆ −1/5 ≥ 10∆ −9/82.5 , namely, ∆ ≤ 10 −11 . By Theorem 2.5, we can then conclude that the truncated EM solutions will approximate the true solution x(t) in the sense that for ∆ ≤ 10 −11 . The problem is that the stepsize needs to be so small that the truncated EM is almost inapplicable. This example shows that condition (2.14) is too restrictive sometimes. Could we remove this condition and still establish the strong convergence theory? We will give our positive answer in the next section. 3 Main Results 3.1 Lemmas. First of all, we modify the choice of function h to make it more flexible by choosing a constantĥ ≥ 1 and a strictly decreasing function h : (3.1) From now on, our function h will satisfy this condition instead of (2.5). There are lots of choices for h(·). For example, h(∆) =ĥ∆ −ϵ for some ϵ ∈ (0, 1/4]. Before we proceed, let us make a useful remark. Remark 3.1. Comparing (3.1) with (2.5), we here simply let ∆ * = 1 and remove condition h(∆ * ) ≥ µ(1) while we also replace condition ∆ 1/4 h(∆) ≤ 1 by a weaker one ∆ 1/4 h(∆) ≤ĥ. In other words, we have made the choice of function h more flexible. We emphasise that such changes do not make any effect on the results in [GMY17]. In fact, condition h(∆ * ) ≥ µ(1) was only used to prove [ [GMY17], Lemmas 2.4 and 4.2]. But, it is easy to show (see the proof of Lemma 3.2 below) that both lemmas there still hold as long as we replace the constant 2K 1 there by and this change does not affect any other results in [GMY17]. It is also easy to check that replacing ∆ 1/4 h(∆) ≤ 1 by ∆ 1/4 h(∆) ≤ĥ does not make any effect on the results in [GMY17].
The following lemma shows that the truncated functions defined by (2.6) preserve the Khasminskii-type condition (2.2) to a very nice degree.
(3.4) Namely, we have showed that the required assertion (3.2) also holds for x ∈ R n with |x| > µ −1 (h(∆)) and any y ∈ R n . The proof is hence complete.
The following lemma shows that the truncated functions f ∆ and g ∆ preserve Assumption 2.4 perfectly. Lemma 3.3. Let Assumption 2.4 hold. Then, for every ∆ ∈ (0, 1], we have This implies the assertion by noting that etc. The proof is complete. Recalling Remark 3.1, we can then cite two lemmas from [GMY17] on the continuous-time truncated EM solutions defined by (2.9) and (2.11) for the use of this paper. From now on we will fix T > 0 arbitrarily and let C stand for generic positive real constants dependent on T, K 1 , K 2 , ξ etc. but independent of ∆ and its values may change between occurrences.
Lemma 3.4. For any ∆ ∈ (0, 1] and any p ≥ 2, we have where c p is a positive constant dependent only on p. Consequently Lemma 3.5. Let Assumption 2.1 hold. Then Lemma 3.4 shows that x ∆ (t) andx ∆ (t) are close to each other in the sense of L p . We also observe thatx ∆ (t) is computable, but x ∆ (t) is not in general. It is thereforē x ∆ (t) that we use in practice. However, for our analysis, it is more convenient to work on both of them. We also emphasize that Lemma 3.4 holds for any p ≥ 2 but Lemma 3.5 holds only for the specifiedp.

Convergence rates.
The following theorem is one of our main results in this paper.
(3.11) By Assumption 2.3, we get where ) ds and ds.
Step 2. Let us now estimate H 1 and H 2 . Noting that

Consequently, we have
( 3.13) To estimate H 2 , we observe that where where t ∧ θ i has been replaced by T as t ∧ θ i ≤ T and the order of integrations has also been exchanged. By Assumption 2.4 and the Hölder inequality as well as Lemma 3.5, we derive that ]) (p−r)/p .

ADVANCES IN TRUNCATED EULER-MARUYAMA METHOD FOR SDDES 11
But Similarly, Consequently, Similarly, we can show We therefore have Let us now estimate H 22 . By Lemma 3.3 and the Hölder inequality as well as Lemma 3.4 and Assumption 2.2, we derive that Similarly, we have Consequently Step 3. Combining (3.12) -(3.16) together yields ) .

WEIYIN FEI, LIANGJIAN HU, XUERONG MAO, DENGFENG XIA
The Gronwall inequality shows Letting i → ∞ gives the first assertion (3.9). The second assertion (3.10) follows from the first one and Lemma 3.4. The proof is therefore complete.
The assertions then follow from Corollary 3.7. It is reasonable to regard the initial data ξ = {ξ(t) : −τ ≤ t ≤ 0} as the observation of the solution on t ∈ [−τ, 0]. Recalling that the Brownian motion is α-Hölder continuous for α ∈ (0, 0.5) (see, e.g., [KR88]), we may assume that Assumption 2.2 holds for some γ ∈ (0, 0.5) close to 0.5. In this case, Corollary 3.8 shows the order of the convergence rate is close to 0.5. This is almost optimal if we recall the order of the classical EM method applied to stochastic differential equations (SDEs) is 0.5 under the global Lipschiz condition (see, e.g., [KP,M97]).

Comparison.
Let us now compare our new Theorem 3.6 with the main result of [GMY17], namely Theorem 2.5 in order to highlight the significant contribution of our new result. Although the assumptions imposed in both theorems are almost the same, we observe the following key differences:

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i). The key feature of Theorem 3.6 is that it does not require the restrictive condition (2.14).
ii). The assertions of Theorem 3.6 hold for any ∆ ∈ (0, 1] while the assertions of Theorem 2.5. hold only for sufficiently small ∆ which satisfies condition (2.14).
iii). Theorem 3.6 needs a slightly stronger condition on the parameters, namelȳ p > r + 2, while Theorem 2.5 needsp > r only. iv). The assertions of Theorem 3.6 look slightly worse than those of Theorem 2.5 but could be the same whenp is sufficiently large, for example, 2ϵ(p−r−2)/(2+r) ≥ 1 − 2ϵ in Corollary 3.7.
The key advantage of our new Theorem 3.6 lies in that it does not need condition (2.14). In Section 2.5, we have shown, via the example, that condition (2.14) could sometimes make Theorem 2.5 inapplicable and hence our new Theorem 3.6 without condition (2.14) is particularly useful in this situation.

Further Results.
In Section 3, we showed that both truncated EM solutions x ∆ (T ) andx ∆ (T ) converge to the true solution x(T ) in L 2 for any T > 0. This is sufficient for some applications e.g. when we need to approximate the European put or call option value at time T (see, e.g., [HM05]). However, we sometimes need to approximate quantities that are path-dependent, for example, the European barrier option value. In these situations, we will need a stronger convergence result like We aim to establish such stronger results in this section. For this purpose, we need to replace Assumption 2.4 with the following slightly stronger one.
Let us make a useful remark.
Remark 5.2. As Assumption 5.1 is stronger than Assumption 2.4 so all the results before hold if Assumption 2.4 is replaced with Assumption 5.1. Moreover, it is easy to see that if Assumption 2.1 holds for somep > 0, then it must hold for anyp > 2 as long as Assumption 5.1 holds as well. In fact, if Assumption 2.1 holds for somē p > 0, then together with (5.2), there holds for any p > 2 for someK 1 . Recalling Corollary 3.8 and its proof, we can then conclude that the term throughout Section 3 can be replaced by if Assumption 2.4 is replaced with Assumption 5.1 and we let µ(u) = K 5 u (2+r)/2 and h(∆) = ∆ −ϵ for some ϵ ∈ (0, 1/4].
We can now state our stronger result under the stronger conditions. Theorem 5.3. Let Assumptions 2.1 -2.3 and 5.1 hold and assume thatp > r + 2.
Let µ(u) = K 5 u (2+r)/2 and h(∆) = ∆ −ϵ for some ϵ ∈ (0, 1/4]. Then, for every ∆ ∈ (0, 1], Proof. We will use the same notation as in the proof of Theorem 3.6. The Itô formula shows that In the same way as in the proof of Theorem 3.6, we can show that Recalling (3.14), we then get By Theorem 3.6 and Remark 5.2 as well as (3.15) and (3.16), we get

ADVANCES IN TRUNCATED EULER-MARUYAMA METHOD FOR SDDES 15
On the other hand, by the Burkholder-Davis-Gundy inequality (see, e.g., [DZ92]), we derive Recalling the estimate on H 2 in the proof of Theorem 3.6 as well as Remark 5.2, we see that Moreover, by Assumption 5.1, It therefore follows from (5.7)that Hence the required assertion (5.4) follows from (5.5) along with (5.6) and (5.8). The proof is complete.
As pointed out in Section 3,x ∆ (t) is computable, but x ∆ (t) is not in general. It would therefore be very useful to have a convergence result like (5.4) but x ∆ (t) there is replaced byx ∆ (t). For this purpose, let us present a lemma.

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Substituting this into (5.11) yields the required assertion (5.9). The proof is complete.
The following more useful theorem follows from Theorem 5.3 and Lemma 5.4 immediately.
We have hence verified Assumption 2.3. It is also straightforward to show that Assumption 2.4 is satisfied with r = 4 (and some K 4 which is not important). To apply Theorem 3.6, we still need to design functions µ and h satisfying (2.4) and (3.1). Note that sup |x|≤u (|f (x)| ∨ |g(x)|) ≤âu 3 , ∀u ≥ 1, whereâ = (|a 1 | + |a 2 | + a 3 ) ∨ (|a 4 | + |a 5 |). We can hence have µ(u) =âu 3 and its inverse function µ −1 (u) = (u/â) 1/3 for u ≥ 0. For ϵ ∈ (0, 1/4], we define h(∆) = ∆ −ϵ for ∆ > 0. By Theorem 3.6, we can then conclude that the truncated EM solutions will converge to the true solution of the SDE (6.4) in the sense that for all ∆ ∈ (0, 1]. In particular, if γ is close to 0.5 (or bigger than half), this shows that the order of convergence is close to 0.5. Example 6.3. Let us still consider the scalar SDDE (6.4) but change the diffusion coefficient into g(x, y) = a 4 x + a 5 y. We see clearly Assumptions 2.1 -2.3 and 5.1 hold. We also let µ(u) =âu 3 for u ≥ 0 and h(∆) = ∆ −ϵ for ϵ ∈ (0, 1/4]. Then, by Theorem 5.5, we can conclude that for every ∆ ∈ (0, 1], (6.6) 7 Conclusion. In this paper, we reviewed one of the main results of [GMY17] and pointed out a restrictive condition imposed there via an example. We then successfully established the strong convergence theory without this restrictive condition. We compared our new result with the one in [GMY17] and highlighted our significant contribution in this paper. We also established a new strong convergence theory for the solutions over a finite time interval, and this was not discussed in [GMY17]. Examples were used to motivate our paper and to illustrate our new theory.