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# Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods

In this paper, a general theorem on the equivalence of pth moment stability between stochastic differential delay equations (SDDEs) and their numerical methods is proved under the assumptions that the numerical methods are strongly convergent and have the bouneded pth moment in the finite time. The truncated Euler-Maruyama (EM) method is studied as an example to illustrate that the theorem indeed covers a large ranges of SDDEs. Alongside the investigation of the truncated EM method, the requirements on the step size of the method are significantly released compared with the work, where the method was initially proposed.

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01/15/2020

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## 1 introduction

Higham, Mao and Stuart in [7] initialised the study on the equivalence of stabilities between solutions of stochastic differential equations (SDEs) and their numerical solutions. To be more precise, it is proved in their paper that underlying solutions are mean square stable if and only if numerical solutions are also stable in the mean square sense. The result applies under the assumption of the finite time convergence of the numerical methods.

Mao in [13] extended such a result to stochastic differential delay equations (SDDEs). Zhao, Song and Liu in [22] investigated such equivalence for SDDEs with Poisson jump and Markov switching. More recently, Liu, Li and Deng in [12] studied neutral delayed stochastic differential equations for such a problem. Deng et la. in [5] extended results in [13] to SDDEs driven by -Brownian motion. All the works mentioned above were devoted to the exponential stability in the mean square sense.

Mao in [14] generalised the results in [7] to the case of pth moments and made some connections of the almost sure stability between SDEs and their numerical solutions. Yang and Li in [19] discussed similar problems in the -framework.

In this paper, we study the equivalence of pth moment stability between solutions of SDDEs and their numerical solutions, which could be regarded as a generalisation of [13]. In addition, we investigate the truncated Euler-Maruyama (EM) method as an example. Compared with those classical Euler-type methods discussed in previous works mentioned above, we do not need to impose the global Lipschitz condition on either the drift or diffusion coefficient.

The truncated EM method was proposed originally by Mao in [15, 16] for SDEs. After that, plenty of works that employed the truncating technique have been done for SDDEs. For example, the truncated EM method was studied by Guo, Mao and Yue in [6], the partially truncated EM method was investigated by Zhang, Song and Liu in [20], and the truncated Milstein method was discussed by Zhang, Yin, Song and Liu in [21].

Our theorem on the truncated EM for SDDEs in this paper is of interest in two aspects. Firstly, the result of the truncated EM demonstrates Theorem 3.3 indeed covers a large class of SDDEs and numerical methods. Secondly, the requirement on the step size of the method is significantly released compared with the existing works, which is a stand-alone interesting result. It should be mentioned that many other interesting numerical methods have been proposed for SDDEs, for example [1, 2, 3, 4, 8, 9, 11, 10, 17, 18, 23] and the references therein.

The main contributions of this paper are summarized as follows.

• A general theorem on the equivalence of pth moment stability between SDDEs and their numerical methods are stated and proved, which covers a large class of SDDEs and various numerical methods.

• The constraint on the step size of truncated EM method for SDDEs is released, which makes the method more applicable.

This paper is constructed in the following way. The mathematical preliminaries are presented in Section 2. Section 3 sees the general theorem of the equivalence of the pth moment stability. Section 4 is devoted to the study on the truncated EM method. Numerical examples are conducted to demonstrate the theoretical results in Section 5. Section 6 concludes this paper by emphasizing the main contributions of this work.

## 2 Mathematical preliminaries

Throughout this paper we use the following notations. Let be the Euclidean norm in and

be the inner product of vectors

. If A is a vector or matrix, its transpose is denoted by . If A is a matrix, its trace norm is denoted by . If is a real number, its integer part is denoted by . Let and . Let denote the family of continuous functions from to .

Let

be a complete probability space with a filtration

satisfying the usual conditions(i.e.,it is right continuous and contains all -null sets). Let be an m-dimensional Brownian motion defined on the probability space. Moreover, for two real numbers a and b, we use and . If G is a set, its indicator function is denoted by , namely if and otherwise. Denote by the family of -measurable,

valued random variables

such that

 ||ξ||p:=sup−τ≤u≤0|ξ(u)|p<∞.

If is a continuous -valued stochastic process on , we let for which is regarded as a valued stochastic process.

Let us consider the n-dimensional autonomous stochastic delay differential equations (SDDEs)

 dy(t)=f(y(t),y(t−τ))dt+g(y(t),y(t−τ))dω(t), (1)

with the initial data , where and .

Denote the numerical solution to the SDDE (1) by , whose detailed structure is not needed in Sections 2 and 3. Our definition of the exponential stability in th moment is as follows.

###### Definition 2.1.

The solution to the SDDE (1) is said to be exponentially stable in th moment for any , if there is a pair of positive constants and such that, for any initial data

 E|y(t;0,ξ)|p≤M||ξ||pe−λt∀t≥0. (2)

In this paper we often need to introduce the solution to the SDDE (1) for initial data give at time t=s. As long as the existence and uniqueness of this solution denoted by on is guaranteed. It is easy to observe that the solutions to the SDDE (1) have the following flow property:

 y(t;0,ξ)=y(t;s,ys)∀0≤s

Moreover, due to the autonomous property of the SDDE (1), the exponential stability (2) implies

 E|y(t;s,ξ)|p≤M||ξ||pe−λ(t−s)∀t≥s.
###### Definition 2.2.

A numerical method to the SDDE (1)is said to be exponentially stable in the moment for any if there is a pair of positive constants and such that with initial data ,

 E|x(t;0,ξ)|p≤H||ξ||pe−γt∀t≥0.

The next two assumptions are needed for Theorem 3.3. Briefly speaking, Assumption 2.3 needs that the underlying and numerical solutions have the finite th moment and Assumption 2.4 requires that the numerical solution converges to the underlying solution in a finite time with any convergence rate.

###### Assumption 2.3.

The underlying solution and the numerical solution to SDDE (1) satisfy

 sup−τ≤t≤τE|y(t;0,ξ)|p≤C∗||ξ||p, (3)

and

 sup−τ≤t≤τE|x(t;0,ξ)|p≤C∗||ξ||p,

respectively, where is a constant independent of .

###### Assumption 2.4.

Write and define which is the numerical solution to the SDDE (1) with initial data starting from , then

 supτ≤t≤τ+TE|x(t)−y(t)|p≤C(T)||ξ||pα(Δ), (4)

where depends on T but not on and and is an increasing function with respect to .

## 3 A general theorem

To prove the main theorem, we present two lemmas firstly.

###### Lemma 3.1.

Let Assumptions 2.3 and 2.4 hold. For any , suppose that the SDDE (1) is exponentially stable in moment, namely

 E|y(t;0,ξ)|p≤M||ξ||pe−λt∀t≥0,

for all . Then there exists a such that for every , the numerical solution to the SDDE (1) is exponentially stable in moment with rate constant and growth constant , both of which are independent of . More precisely,

 E|x(t;0,ξ)|p≤H||ξ||pe−γt∀t≥0,

with and , where

 T=τ(9+In[4log(2pM)/λτ]).
###### Proof.

Fix any initial data , write and define . The exponential stability in the moment if the SDDE (1) shows

 E|y(t)|p≤M||xτ||pe−τ(t−τ)for ∀t≥τ. (5)

By the definition of T, we observe that

 2pMe−λ(T−2τ)≤e−34λT. (6)

By the elementary inequality

 (a+b)p≤(2(a∨b)p)≤2p(ap∨bp)≤2p(ap+bp)∀a,b≥0,

we have

 E|x(t)|p≤2p(E|x(t)|−E|y(t)|p+E|y(t)|p). (7)

By (4) and (5)

 supT−τ≤t≤2T−τE|x(t)|p≤2p(C(2T−2τ)||ξ||pα(Δ)+M||xτ||pe−λ(T−2τ))≤[2pC(2T−2τ)α(Δ)+2pMe−λ(T−2τ)]sup−τ≤t≤τE|x(t)|p,

if necessary, let be even smaller so that

 2pC(2T−2τ)α(Δ)+2pMe−λ(T−2τ)≤e(−12λT).

Then

 supT−τ≤t≤2T−τE|x(t)|p≤e−12λTsup−τ≤t≤τE|x(t)|p. (8)

Recall that T is a multiple of and hence of . So, by the flow property, for any integer

 x(t)=x(t;iT,xiT)∀t≥iT.

Repeating the argument above for in the same way that (8) was obtained we may establish

 sup(i+1)T−τ≤t≤(i+2)T−τE|x(t)|p≤e−12λTsupiT−τ≤t≤iT+τE|x(t)|p.

From this we see that

 sup(i+1)T−τ≤t≤(i+2)T−τE|x(t)|p≤e−12λTsupiT−τ≤t≤(i+1)T−τE|x(t)|p.

By iteration, we can get that

 sup(i+1)T−τ≤t≤(i+2)T−τE|x(t)|p≤e−12λ(i+1)Tsup−τ≤t≤T−τE|x(t)|p. (9)

Now, by (4) and (5), we see from (7) that

 supτ≤t≤T−τE|x(t)|p≤2p(E|x(t)−y(t)|p+E|y(t)|p)≤2p(C(T−2τ)||ξ||Pα(Δ)+M||xτ||p)≤(2pC(T−2τ)α(Δ)+2pM))sup−τ≤t≤τE|x(t)|p.

Choose then

 2pC(T−2τ)α(Δ)+2pM≤2p+1M.

Then

 supτ≤t≤T−τE|x(t)|p≤2p+1Msup−τ≤t≤τE|x(t)|p.

Substituting this into (9) and bearing in mind that M must not be less than 1 we obtain that

 sup(i+1)T−τ≤t≤(i+2)T−τE|x(t)|p≤2p+1Me−12λ(i+1)Tsup−τ≤t≤τE|x(t)|p≤2p+1MC∗||ξ||pe−12λ(i+1)T∀i≥0,

while

 sup0≤t≤T−τE|x(t)|p≤2p+1MC∗||ξ||p.

Hence

 E|x(t)|p≤2p+1MC∗e12λT||ξ||pe−12λt∀t≥0.

That is, the numerical method is exponentially stable in the pth moment with and . This completes the proof. ∎

Lemma 3.1 shows that the expenential stable in the pth moment of the SDDE(1) implies the exponential stability in the pth moment of the numerical method for small . Let us now eatablish the converse theorem.

###### Lemma 3.2.

Let Assumption 2.4 hold. Assume that for some , the numerical method is exponentially stable in the pth moment on the SDDE (1), namely

 E|x(t;0,ξ)|p≤H||ξ||pe−γt∀t≥0,

for all . Suppose we can verify

 2pC(2T−2τ)α(Δ)+2pHe−γ(T−2τ)≤e(−12λT), (10)

where . Then the SDDE is exponentially stable in the pth moment. More precisely,

 E|y(t;0,ξ)|p≤ME||ξ||pe−λt∀t≥0,

with

 λ=12γ  and  M=2p+1HC∗e12γT.

The proof of this lemma is proved in the same way as lemma 3.1 was proved, so we only give the outline but highlight the different part.

###### Proof.

Fix any initial data , write and define . The exponential stability in the pth moment of the numerical method shows

 E|x(t)|p≤H||yτ||pe−γ(t−τ)for ∀T≥τ. (11)

By (4) and (11) we can show that

 E|y(t)|p≤2p(E|x(t)−y(t)|p+E|x(t)|p)supT−τ≤t≤2T−τE|y(t)|p≤2p(C(2T−2τ)||ξ||pα(Δ)+H||yτ||pe−γ(T−2τ)).

Using (10) we get that

 supT−2τ≤t≤2T−τE|y(t)|p≤e−12γTsup−τ≤t≤τE|y(t)|p.

Repeating thie argument we find that

 sup(i+1)T−τ≤t≤(i+2)T−τE|y(t)|p≤e−12γ(i+1)Tsup−τ≤t≤T−τE|y(t)|p. (12)

On the other hand,by (4) and (11), we can show that

 supτ≤t≤T−τE|y(t)|p≤(2pC(T−2τ)α(Δ)+2pHsup−τ≤t≤τE|y(t)|p.

This together with (3), yields

 sup−τ≤t≤T−τE|y(t)|p≤C∗(2pC(T−2τ)α(Δ)+2pH)||ξ||p. (13)

It then follows from (12) and (13) that

 E|y(t)|p≤C∗2p+1He12γTe−12γt∀t≥0,

as required. ∎

Now, we are ready for the main theorem.

###### Theorem 3.3.

Under Assumption 2.3 and Assumption 2.4, the SDDE (1) is exponentially stable in pth moment if and only if for some , the numerical method is exponentially stable in pth moment.

Combining lemmas 3.1 and 3.2 we obtain the following sufficient and necessary theorem.

###### Remark 3.4.

The reason that we regard Theorem 3.3 as a general result is due to the assumptions we made, where only the finite time convergence and the moment boundedness of the numerical method in a very short time are needed, but no particular structure of the method is specified.

## 4 The truncated EM method

This section is to show that the truncated EM method is a numerical approximation whose th moment exponential stability is equivalent to that of the underlying SDDEs. To achieve this, we show the finite time strong convergence as well as the moment boundedness of the method firstly. Then the application of Theorem 3.3 implies the desired result.

### 4.1 Brief introduction

To make this paper self contained, we brief the truncated EM method for SDDEs in this part, along which we also discuss the fact that the requirement on the step size is weaken in this paper.

For the SDDE 1, we impose following assumptions on the drift and diffusion coefficients.

###### Assumption 4.1.

Assume that the coefficients f and g satisfy the local Lipschitz condition: For any , there is a such that

 |f(x,y)−f(¯x,¯y)|∨|g(x,y)−g(¯x,¯y)|≤KR(|x−¯x|+|y−¯y|)|x|∨|y|∨|¯x|∨|¯y|≤R,

for all

###### Assumption 4.2.

Assume that the coefficients satisfy the Khasminskii-type condition: There is a pair of constants and such that

 xTf(x,y)+p−12|g(x,y)|2≤K1(1+|x|2+|y|2),

for all

###### Assumption 4.3.

We need an additional condition. To state it, we need a new notation. Let denote the family of continuous functions such that for each there is a positive constant for which

 U(x,¯x)≤Kb|x−¯x|2∀x,¯x∈Rn, with |x|∨|¯x|≤b.

Assume that there is a pair of constants and such that

 (x−¯x)T(f(x,y)−f(¯x,¯y))+q−12|g(x,y)−g(¯x,¯y)|2≤H1(|x−¯x|2+|y−¯y|2)−U(x,¯x)+U(y,¯y),

for all

###### Assumption 4.4.

Assume that there is a pair of positive constants and such that

 |f(x,y)−f(¯x,¯y)|2∨|g(x,y)−g(¯x,¯y)|2≤H2(1+|x|ρ+|y|ρ+|¯x|ρ+|¯y|ρ)(|x−¯x|2+|y−¯y|2),

for all

###### Assumption 4.5.

There is a pair of constants and such that the initial data satisfies

 |ξ(u)−ξ(v)|≤K2|u−v|γ,−τ≤v

To define the truncated EM numerical solutions, we first choose a strictly increasing continuous function such that as and

 sup|x|∨|y|≤r(|f(x,y)|∨|g(x,y)|)≤μ(r),∀r≥1.

Denoted by is the inverse function of and we see that is a strictly increasing continuous function from to . We also choose a constant and a strictly decreasing function such that

 limΔ→0h(Δ)=∞  and  Δ1/4h(Δ)≤^h,∀Δ∈(0,1].

We will later that Assumption 4.4 implies (17), namely that both coefficients f and g grow at most polynomially, hence we can let , where is a positive constant specified in (17). Moreover, we can let for some . In other words, there are lots of choices for and .

A comparison of assumptions of the method in this paper and those in the previous work is presented in Appendix Appendix.

For a given step size , let us define a mapping from to the closed ball by

 πΔ(x)=(|x|∧μ−1(h(Δ)))x|x|,

where we set when . That is, will map to itself when and to when . We then define the truncated functions

 fΔ(x,y)=f(πΔ(x),πΔ(y))  and  gΔ(x,y)=g(πΔ(x),πΔ(y)),

for . It is easy to see that

 |fΔ(x,y)|∨|gΔ(x,y)|≤μ(μ−1(h(Δ)))=h(Δ),∀x,y∈Rn. (14)

From now on, we will let the step size be a fraction of That is, we will use for some positive integer M. When we use the terms of a sufficiently small we mean that we choose M sufficiently large.

Let us now form the discrete-time truncated EM solutions. Define for Set for and then form

 Xk+1=Xk+fΔ(Xk,Xk−M)Δ+gΔ(Xk,Xk−M)ΔWk,

for where In our analysis, it is more convenient to work on the continuous-time step process on defined by

 ¯x(t)=∞∑k=−MXk1[kΔ,(k+1)Δ)(t),

where is the indicator function of (please recall the notation defined in the beginning of this paper). The other one is the continuous-time continuous process on defined by for while for

 x(t)=ξ(0)+∫t0fΔ(¯x(s),¯x(s−τ))ds+∫t0gΔ(¯x(s),¯x(s−τ))dW(s).

We see that is an Itô process on with its Itô differential

 dx(t)=fΔ(¯x(s),¯x(s−τ))dt+gΔ(¯x(s),¯x(s−τ))dW(t). (15)

It is useful to know that for every , namely they coincide at Of course, is computable but is not in general.

### 4.2 Results

###### Lemma 4.6.

Let Assumption 4.2 hold. Then, for all , we have

 xTfΔ(x,y)+p−12|gΔ(x,y)|2≤^k(1+|x|2+|y|2)∀x,y∈Rn, (16)

where .

###### Proof.

Fix any For with and any ,by Assumption 4.2 we have,

 xTfΔ(x,y)+p−12|gΔ(x,y)|2=πΔ(x)Tf(πΔ(x),πΔ(y))+p−12|g(πΔ(x),πΔ(y))|2≤K1(1+|πΔ(x)|2+|πΔ(y)|2)≤K1(1+|x|2+|y|2),

which implies the desired assertion (16). On the other hand, for with and any by Assumption 4.2, we have

 xTfΔ(x,y)+p−12|gΔ(x,y)|2=πΔ(x)Tf(πΔ(x),πΔ(y))+p−12|g(πΔ(x),πΔ(y))|2+(x−πΔ(x))Tf(πΔ(x),πΔ(y))≤K1(1+|πΔ(x)|2+|πΔ(y)|2)+(x−πΔ(x))Tf(πΔ(x),πΔ(y))≤K1(1+|πΔ(x)|2+|πΔ(y)|2)+(|x|μ−1(h(Δ))−1)πΔ(x)Tf(πΔ(x),πΔ(y))≤|x|μ−1(h(Δ))K1(1+|πΔ(x)|2+|πΔ(y)|2)≤K1(1∨1μ−1(h(Δ)))(|x|+|x|2+|x||y|),

by the element inequality , we can get that

 K1(1∨1μ−1(h(Δ)))(|x|+|x|2+|x||y|)≤2K1(1∨1μ−1(h(Δ)))(1+|x|2+|y|2)≤^k(1+|x|2+|y|2),

where

The following lemma is useful to observe that the truncated functions and preserve assumption 4.4 perfectly.

###### Lemma 4.7.

Let Assumption 4.4 hold. Then, for all we have

 |fΔ(x,y)−fΔ(¯x,¯y)|2∨|gΔ(x,y)−gΔ(¯x,¯y)|2≤H2(1+|x|ρ+|y|ρ+|¯x|ρ+|¯y|ρ)(|x−¯x|2+|y−¯y|2),

for all

###### Proof.
 |fΔ(x,y)−fΔ(¯x,¯y)|2∨|gΔ(x,y)−gΔ(¯x,¯y)|2=|f(πΔ(x),πΔ(y))−f(πΔ(¯x),πΔ(¯y))|2∨|g(πΔ(x),πΔ(y))−g(πΔ(¯x),πΔ(¯y))|2≤H2(1+|πΔ(x)|ρ+|πΔ(y)|ρ+|πΔ(¯x)|ρ+|πΔ(¯y)|ρ)(|πΔ(x)−πΔ(¯x)|2+|πΔ(y)−πΔ(