ALMOST SURE EXPONENTIAL STABILITY IN THE NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

. This paper is mainly concerned with whether the almost sure exponential stability of stochastic diﬀerential equations (SDEs) is shared with that of a numerical method. Under the global Lipschitz condition, we ﬁrst show that the SDE is p th moment exponentially stable (for p ∈ (0 , 1)) if and only if the stochastic theta method is p th moment exponentially stable for a suﬃciently small step size. We then show that the p th moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively. Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method, instead of the method of the Lyapunov functions. That is, we can now carry out careful numerical simulations using the stochastic theta method with a suﬃciently small step size Δ t . If the stochastic theta method is p th moment exponentially stable for a suﬃciently small p ∈ (0 , 1), we can then infer that the underlying SDE is almost surely exponentially stable. Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs. In particular, we give positive answers to two open problems, (P1) and (P2) listed in section 1.


Introduction. Stochastic differential equations (SDEs) have been widely
used in many branches of science and industry with the emphasis being on stability analysis [1,6,11,13,16].One of the most powerful techniques in the study of stochastic stability (e.g., the moment exponential stability or almost sure exponential stability) is the method of Lyapunov functions.Suppose that we are required to find out whether an SDE is stochastically stable.In the absence of an appropriate Lyapunov function, we may carry out careful numerical simulations using a numerical method with a "small" step size Δt.We are then left with two key questions: (Q1) If the SDE is stochastically stable (e.g., mean square exponentially stable or almost surely exponentially stable), will the numerical method be stochastically stable?(Q2) If the numerical method is stochastically stable for small Δt, can we infer that the underlying SDE is stochastically stable?There are various types of stochastic stabilities, including the exponential stability in mean square, almost sure exponential stability, and asymptotic stability in probability [11,14,15].In any case, both questions (Q1) and (Q2) deal with asymptotic (t → ∞) properties and hence cannot be answered directly by applying traditional finite-time convergence results.
In the case where stochastic stability means exponential stability in mean square, results that answer (Q1) and (Q2) for scalar, linear systems can be found in [7,20,21].Baker and Buckwar [3] consider the stability of numerical methods for scalar constant delay SDEs under the assumptions of the global Lipschitz coefficients and the existence of a Lyapunov function.Schurz [21] also has results for nonlinear SDEs.In particular, for nonlinear SDEs under the global Lipschitz condition, Higham, Mao, and Stuart [9] show that the exponential stability in mean square for the SDE is equivalent to the exponential stability in mean square of the numerical method (e.g., the Euler-Maruyama and the stochastic theta method) for sufficiently small step sizes.For further developments in this area, we refer the reader to [5,8,19,22,24,27], for example, and the references therein.
In the case where stochastic stability means almost sure exponential stability, so far most results answer (Q1), but few address (Q2).In fact, unlike in the mean square case, the first paper that addressed these questions on a reasonable class of SDEs was Higham, Mao, and Stuart [10] in 2007.In their paper, they answered (Q1) and (Q2) for the linear scalar SDE using the Euler-Maruyama method.For the nonlinear SDE, they answered (Q1) using the Euler-Maruyama method for the SDE under the linear growth condition plus the additional condition which guaranteed the almost sure exponential stability of the SDE.For the nonlinear SDE without the linear growth condition, they answered (Q1) using the backward Euler method.The research in this area has since then been developed by many authors, e.g., [4,17,25,26], but all of these authors have addressed (Q1).There are many open problems in this direction.Two of them are stated below: (P1) If the multidimensional linear SDE (1.1) is almost surely exponentially stable, will a numerical method be almost surely exponentially stable for all sufficiently small step sizes?(P2) Under the conditions in Theorem 4.5 below, the nonlinear SDE (2.1) is almost surely exponentially stable.This theorem is one of the most useful criteria on almost sure exponential stability.However, will a numerical method be almost surely exponentially stable for all sufficiently small step sizes?Should we have some answers to (Q1) and (Q2) in the case where stochastic stability means almost sure exponential stability, these problems would be solved to a certain degree.In this paper we will address (Q1) and (Q2) in the sense of almost sure exponential stability in two steps.Under the global Lipschitz condition, we first show that the SDE is pth moment exponentially stable (for p ∈ (0, 1)) if and only if the stochastic theta method is pth moment exponentially stable for a sufficiently small step size.We will then show that the pth moment exponential stability of the SDE or the stochastic theta method implies the almost sure exponential stability of the SDE or the stochastic theta method, respectively.Hence, our new theory enables us to study the almost sure exponential stability of the SDEs using the stochastic theta method instead of the method of the Lyapunov functions.That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt.If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable.Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs.In particular, we will be able to solve problems (P1) and (P2).In fact, it is known (see, e.g., [2]) that the linear SDE (1.1) is almost surely exponentially stable if and only if it is pth moment exponentially stable for a sufficiently small p ∈ (0, 1).Moreover, under the conditions of Theorem 4.5, we will show that the nonlinear SDE (2.1) is pth moment exponentially stable for a sufficiently small p ∈ (0, 1).Applying our new results, we will hence have the positive results on both (P1) and (P2).
Before we proceed to establish our new theory, we should point out that the way we establish our new results on (Q1) and (Q2) in the sense of the pth moment exponential stability for sufficiently small p ∈ (0, 1) is motivated by our earlier paper [9] that deals with the 2nd moment exponential stability.However, there are at least two significant differences which we highlight below: • It was assumed in [9] that a numerical method is available which, given a step size Δt > 0, computes discrete approximations x k ≈ y(kΔt), with x 0 = y 0 .It was also assumed that there is a well-defined interpolation process that extends the discrete approximation {x k } k∈Z + to a continuous time approximation {x(t)} t∈R + , with x(kΔt) = x k .However, in general, only the discrete approximations x k are computable but not the continuous-time approximation {x(t)} t∈R + .It is therefore more useful in practice if the theory is only based on the discrete approximations x k , and this is what we will establish in this paper.In other words, in this paper, we only need a numerical method which computes the discrete approximations x k ≈ y(kΔt), and we do not require the continuous-time approximations.• Mathematically speaking, many inequalities used in [9] for the 2nd moment do not work for the pth moment when p ∈ (0, 1).We therefore have to develop new techniques to handle the pth moment.Moreover, our new theory is based on the discrete approximations x k , so we have to apply the discretetime analysis in many proofs in this paper, while [9] used the continuous-time analysis essentially.

Stochastic differential equations.
Throughout this paper, we let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions (i.e., it is right continuous and increasing while F 0 contains all P-null sets).Let w(t) = (w 1 (t), . . ., w m (t)) T be an m-dimensional Brownian motion defined on the probability space.Let  Moreover, f (0) = 0 and g(0) = 0. We should point out that the reason we assume that f (0) = 0 and g(0) = 0 is because this paper is concerned with the stochastic stability of the trivial solution y(t) ≡ 0. It is also easy to see that this assumption implies the linear growth condition Moreover, it is well known (see, e.g., [1,6,16]) that under Assumption 2. This, by the Gronwall inequality, yields Hence, when p ∈ (0, 1), In other words, we have for p ∈ (0, 1).
Of course, we may consider a more general case, for example, where the SDE has its random initial data y(0) = ξ which is an F 0 -measurable R n -valued random variable such that E|ξ| p < ∞ ∀p > 0. In this case, by the Markov property of the solution, we can easily see that the solution satisfies ( for any p > 0. In this section we consider the pth moment exponential stability of the origin, which we define as follows (see, e.g., [6,11,14,15]).Definition 2.2.Let p > 0. The SDE (2.1) is said to be exponentially stable in the pth moment if there is a pair of positive constants λ and M such that, for every initial value y 0 ∈ R n , (2.9) We refer to λ as a rate constant and M as a growth constant.
As in the explanation of (2.8) we see that (2.9) is equivalent to the following more general form: (2.10) Downloaded 03/20/20 to 80.6.87.97.Redistribution subject to CCBY license 2.2.Numerical solutions.We suppose that a numerical method is available which, given a step size Δt > 0, computes discrete approximations x k ≈ y(kΔt) for k ∈ Z + with x 0 = y 0 .We also require that the process defined by the numerical method possess the following Markov property: • Given xk for some k ∈ Z + , the process {x k } k≥ k can be regarded as the process which is produced by the numerical method applied to the SDE (2.1) on t ≥ kΔt with the initial data y( kΔt) = xk.Hence, the probability distributions of the process {x k } k≥ k are fully determined, given xk, but how the process has reached xk has no further use.In other words, the discrete-time process {x k } k∈Z + is a Markov process.Moreover, by the time-homogeneity of the SDE, {x k } k∈Z + is of course time-homogeneous.Such a discrete-time process will be illustrated for the class of stochastic theta methods in the next section.Following Definition 2.2, we now define the pth moment exponential stability for the discrete-time approximate solutions {x k } k∈Z + .
Definition 2.3.Let p > 0. For a given step size Δt > 0, a numerical method is said to be exponentially stable in the pth moment on the SDE (2.1) if there is a pair of positive constants γ and N such that with initial value By the time-homogeneous Markov property, we see that (2.11) is equivalent to the following more general form: (2.12)

Assumptions and results
. From now on we will let p ∈ (0, 1).We wish to know whether the numerical method shares the pth moment exponential stability with the SDE.That is, we wish to address both (Q1) and (Q2) in the sense of the pth moment exponential stability.For this purpose, we impose a natural finite pth moment condition on the numerical methods.
Assumption 2.4.Let p ∈ (0, 1).For all sufficiently small Δt the numerical method applied to (2.1) with initial condition where H(T, p, K) is a positive constant dependent on T, p, K only, but independent of the initial value y 0 and the step size Δt.By the time-homogeneous Markov property of the numerical method, we see easily from this assumption that We also impose a natural finite-time convergence condition on the numerical methods.
Assumption 2.5.Let p ∈ (0, 1).For all sufficiently small Δt the numerical method applied to (2.1) with initial condition where C T depends on T but not on y 0 and Δt, and h : R + → R + is a strictly increasing continuous function with h(0) = 0.
Our notation emphasizes the dependence of C upon T as this is important in the subsequent analysis.It is also easy to see that C T is nondecreasing in T .Moreover, for any k ∈ Z + , if we let ŷ(t) be the solution of the SDE (2.1) on t ≥ kΔt with initial data ŷ( kΔt) = xk, then, by the time-homogeneity of the SDE (2.1), condition (2.15) implies (2.16) sup These assumptions will be illustrated for the class of stochastic theta methods in the next section.The following lemma gives a positive answer to question (Q1) from section 1.
(Please note that both γ and N are independent of Δt.) Proof.Without loss of any generality, we let Δt < 1.We divide the whole proof into 2 steps.
Step 1.By the definition of T , we observe that (2.17) By the elementary inequality In particular, for kΔt ∈ [T, T + 1] (such k exists as Δt < 1), using conditions (2.15) and (2.9), we then have This, together with (2.17), yields Choose Δt ∈ (0, 1) sufficiently small for Then, for every 0 where N = H(T + 1, p, K)e Repeating this procedure, we can show that for any nonnegative integer i, Consequently, and then That is, the numerical method is exponentially stable in the pth moment on the SDE (2.1) with rate constant γ = 1 2 λ and growth constant N = H(T + 1, p, K)e Using Assumptions 2.4 and 2.5 and (2.11), we obtain where M = H(T, p, K)e 1 2 γT .Let us now consider the solution y(t) on t ≥ T .As explained before, this can be regarded as the solution of the SDE (2.1) with the initial data y(T ) at time t = T .Moreover, let {x k } k≥ k be the process which is produced by the numerical method applied to the SDE (2.1) on t ≥ T with the initial data xk = y(T ).By the timehomogeneity of the SDE (2.1) as well as the Markov property of the true and numerical solutions, condition ( Consequently, and then That is, the SDE (2.1) is pth moment exponentially stable with rate constant λ = 1 2 γ and growth constant M = H(T, p, K)e 1 2 γT .The proof is hence complete.Lemmas 2.6 and 2.7 lead to the following theorem.Theorem 2.8.Suppose that a numerical method satisfies Assumptions 2.4 and 2.5.Then the SDE is exponentially stable in the pth moment if and only if there exists a Δt > 0 such that the numerical method is exponentially stable in the pth moment with rate constant γ, growth constant N , step size Δt, and global error constant C T for T = kΔt satisfying (2.30), where k is the smallest integer which is no less than 4 log(2 p N )/(γΔt).
Proof.The "if" part of the theorem follows from Lemma 2.7 directly.To prove the "only if" part, suppose the SDE is exponentially stable in the pth moment with rate constant λ and growth constant M .Lemma 2.6 shows that there is a Δt > 0 such that for any step size 0 < Δt ≤ Δt , the numerical method is exponentially stable in the pth moment with rate constant γ = 1 2 λ and growth constant N = 2 p (C T +1 + M )e 1 2 λ(T +1) , where T = 3+(4 log(2 p M ))/λ.Noting that both of these constants are independent of Δt, it follows that we may reduce Δt if necessary until (2.30) becomes satisfied.
We emphasize that Theorem 2.8 is an "if and only if" result, which shows that, under Assumptions 2.4 and 2.5 and for sufficiently small Δt, the pth moment exponential stability of the method is equivalent to the pth moment exponential stability of the SDE.Thus it is feasible to investigate exponential stability of the SDE from careful numerical simulations.

Improved results.
In Lemmas 2.6 and 2.7, we found new rate constants that were within a factor 1  2 of the given ones.We can of course make the new rate constants as close as possible to the given ones, say a factor of 1 − for any ∈ (0, 1).The price we paid for this was an increase in the growth constants and a decrease in the step sizes.The following lemmas describe this situation more precisely.
For kΔt ∈ [T, T + 1], it then follows from (2.20) that Choose Δt ∈ (0, 1) sufficiently small for Then, for every 0 Let where N = 2 p (C T +1 + M )e (1− )λ(T +1) .The remaining proof is almost the same as Step 2 in the proof of Lemma 2.6.Lemma 2.10.Let ∈ (0, 1) and assume that Assumptions 2.4 and 2.5 hold.Assume also that for a step size Δt > 0, the numerical method is pth moment exponentially stable with rate constant γ and growth constant N .If Δt satisfies where T = kΔt and k is the smallest integer which is not less than 2 log(2 p N )/( γΔt), then the SDE (2.1) is pth moment exponentially stable with rate constant λ = (1− )γ and growth constant M = H(T, p, K)e (1− )γT , where H(T, p, K) is given in (2.4).The proof is similar to that of Lemma 2.7 and so is omitted.

3.
The stochastic theta method.The theory established in the previous section requires that the numerical solutions not only have the Markov property described in section 2 but also that they satisfy Assumptions 2.4 and 2.5.The question is, do such numerical solutions exist?We will give a positive answer here by considering the class of stochastic theta methods.Given a free parameter θ ∈ [0, 1], the numerical solutions by the stochastic theta method are defined by (3.1) with the initial value x 0 = y 0 , where Δw k = w((k + 1)Δt) − w(kΔt).When θ = 0, (3.1) is the widely used Euler-Maruyama method (see, e.g., [12,16]).In this case, (3.1) is an explicit equation that defines x k+1 .But when θ = 0, (3.1) represents a nonlinear system that is to be solved for x k+1 .By the classical Banach fixed-point theorem, it is easy to show the following (see, e.g., [9,23]): • Under the global Lipschitz condition (2.1), if KθΔt < 1, then (3.1) can be solved uniquely for x k+1 , with probability 1.From now on we always assume that the step size Δt < 1/(Kθ), so that the stochastic theta method (3.1) is well-defined.(We will in fact require Δt < 1/( √ 10Kθ) later.)In other words, we can compute the discrete approximations x k ≈ x(kΔt), with x 0 = y 0 .
For our theory to work, we first need to show that the stochastic theta method possesses the Markov property, namely, the discrete process {x k } k∈Z + is a timehomogeneous Markov process.This can been seen easily because, given xk for some k ∈ Z + , the process {x k } k≥ k can be fully determined by (3.1), but how the process has reached xk has no further use.The following theorem shows that the stochastic theta method satisfies Assumption 2.4.
Theorem 3.1.Let Assumption 2.1 hold.Let p ∈ (0, 1) and let Δt be sufficiently small for √ 10KθΔt < 1.Then the discrete process {x k } k∈Z + defined by the stochastic theta method (3.1) satisfies where H(T, p, K) = ( 11) p/2 e 5pT K 2 (T +1) .Proof.To prove the lemma, let us introduce two continuous-time step processes, x k+1 1 [kΔt,(k+1)Δt) (t), with 1 G denoting the indicator function for the set G. It is easy to see that z 1 (kΔt) = z 2 ((k − 1)Δt) = x k .For convenience, we will let t k = kΔt for k ∈ Z + from now on.It is easily shown that Noting that Hence By the linear growth condition (2.3) (followed from Assumption 2.1) as well as the Hölder inequality and the property of the Itô integral, we can show This, together with the condition KθΔt < 1/ √ 10, yields By the discrete Gronwall inequality (see, e.g., [14,15]), we hence obtain sup Finally, we have sup as required.
Let us now proceed to show that the stochastic theta method satisfies Assumption 2.5.We need a lemma.Lemma 3.2.Let Assumption 2.1 hold.Let Δt be sufficiently small for 2K 2 Δt < 1.Then the solution of the SDE (2.1) has the property we can show easily that By (2.5), we then have The proof is therefore complete.Theorem 3.3.Let Assumption 2.1 hold.Let p ∈ (0, 1) and let Δt be sufficiently small for Then the stochastic theta method solution (3.1) and the true solution of the SDE (2.1) satisfy where which is independent of Δt and y 0 .Proof.It follows from (2.1) and (3.3) that for any 0 ≤ t k+1 ≤ T , Define Then Hence Almost sure exponential stability.Our paper is mainly concerned with the almost sure exponential stability of both true and numerical solutions with the objective of finding positive answers to problems (P1) and (P2) in section 1.It is therefore time to relate the pth moment exponential stability to the almost sure exponential stability.We first cite a theorem from [18, Theorem 5.9, p. 167] which shows that under our standing hypothesis, the pth moment exponential stability of the true solutions implies the almost sure exponential stability. Theorem That is, the method is also almost surely exponentially stable.
Proof.Let ∈ (0, γ) be arbitrary.By the Chebyshev inequality, By the well-known Borel-Cantelli lemma, we see that for almost all ω ∈ Ω, holds for all but finitely many k.Hence, there exists a k 0 (ω) ∀ω ∈ Ω excluding a P-null set, for which (4.2) holds whenever k ≥ k 0 .Consequently, for almost all ω ∈ Ω, if kΔt ≤ t ≤ (k + 1)Δt and k ≥ k 0 , ) .• Suppose that we are required to find out whether the SDE (2.1) is almost surely exponentially stable.In the absence of an appropriate Lyapunov function, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt.If the stochastic theta method is pth moment exponential stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable.

Hence lim sup
• If the SDE is pth moment exponentially stable for some p ∈ (0, 1), then the stochastic theta method is almost surely exponentially stable for all sufficiently small step sizes Δt.In other words, we have obtained positive answers to both (Q1) and (Q2) in section 1.We are also in a position to give our positive answers to problems (P1) and (P2) of that section.From this theorem and our theory in sections 2 and 3, we obtain the following theorem, which gives a positive answer to problem (P1).

Answer to (P1
Theorem 4.4.If the linear SDE (4.3) is almost surely exponentially stable, then the stochastic theta method is almost surely exponentially stable for all sufficiently small step sizes.
Theorem 4.5.Let Assumption 2.1 hold.Assume that there exists a function that for all y = 0 and t ≥ 0, where the diffusion operator L acting on the C 2,1 -functions is defined by Then the SDE (2.1) is almost surely exponentially stable.
The following lemma shows that under the same conditions as those of Theorem 4.5, the SDE (2.1) is pth moment exponentially stable for all sufficiently small p.
An application of the Itô formula implies e λt EU (y(t), t) ≤ U (y 0 , 0) ∀t ≥ 0. This yields the required assertion (4.5) by the fact c From this lemma and our theory in sections 2 and 3, we obtain the following theorem, which gives a positive answer to problem (P2).
Theorem 4.7.Under the conditions of Theorem 4.5, the stochastic theta method is almost surely exponentially stable for all sufficiently small step sizes.

Examples.
In this section we just discuss two examples to illustrate our positive answers to (P1) and (P2).
Example 5.1.Consider a two-dimensional linear SDE dy(t) = A 0 y(t)dt + A 1 y(t)dw 1 (t) + A 2 y(t)dw 2 (t) (5.1) on t ≥ 0 with initial value y(0) = y 0 ∈ R 2 , where where I 2 is the 2 × 2 identity matrix.It is therefore easy to see that the linear SDE (5.1) is almost surely exponentially stable if and only if a < 0.5(σ 2 1 − σ 2 2 ).By Theorem 4.4, we can then conclude that the stochastic theta method applied to the SDE (5.1) is almost surely exponentially stable for all sufficiently small step sizes, which is one of the key results of [4].Figure 1 shows a computer simulation of the paths of y 1 (t) and y 2 (t) using the stochastic theta method with the parameter θ = 0.5 and the step size Δt = 0.001 and the system parameters a = 1, σ 1 = 2, σ 2 = 1 as well as the initial values y 1 (0) = 1, y 2 (0) = 2.The computer simulation clearly supports our theoretical result.By Theorem 4.5, the SDE (5.3) is almost surely exponentially stable, while by Theorem 4.7, the stochastic theta method applied to the SDE is also almost surely exponentially stable for all sufficiently small step sizes.Figure 2 is the computer simulation of the paths of y 1 (t) and y 2 (t) using the Euler-Maruyama method (i.e., the stochastic theta method with the parameter θ = 0) and the step size Δt = 0.001 as well as the initial values y 1 (0) = 1, y 2 (0) = 2. Again, the computer simulation clearly supports our theoretical result.(C) denotes the pth moment exponential stability of the stochastic theta method for a sufficiently small step size, (D) denotes the almost sure exponential stability of the stochastic theta method for a sufficiently small step size.In particular, our new theory enables us to study the almost sure exponential stability of SDEs using the stochastic theta method, instead of the method of the Lyapunov functions.That is, we can now carry out careful numerical simulations using the stochastic theta method with a sufficiently small step size Δt.If the stochastic theta method is pth moment exponentially stable for a sufficiently small p ∈ (0, 1), we can then infer that the underlying SDE is almost surely exponentially stable.
Our new theory also enables us to show the ability of the stochastic theta method to reproduce the almost sure exponential stability of the SDEs.In the case when the underlying SDE is linear (namely, (4.3)), recalling the classical result (A) ⇔ (B) established by Arnold, Kliemann, and Oeljeklaus [2], we have shown (A) ⇒ (D).That is, we have obtained the positive answer to problem (P1).In the case when the underlying SDE is nonlinear, we know that (A) does not imply (B) in general.However, we have recalled one of the most useful criteria, Theorem 4.5, for almost sure exponential stability and shown that the SDE is pth moment exponentially stable for all sufficiently small p under the same conditions.Consequently, we have obtained a positive answer to problem (P2).

c
2015 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license Downloaded 03/20/20 to 80.6.87.97.Redistribution subject to CCBY license
(4.1) follows by letting → 0. Before we proceed to give our positive answers to problems (P1) and (P2), let us highlight what our new theory established thus far enables us to do under the standing Assumption 2.1:

| • | denote both the Euclidean norm in R n and the trace norm in R n×m . For a, b ∈ R, we use a ∨ b and a ∧ b for max{a, b} and min{a, b},
|f (x) − f (y)| ∨ |g(x) − g(y)| ≤ K|x − y| ∀x, y ∈ R n .c2015 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license Step 2. Let us now consider the approximate solutions x k for k ≥ k.By the Markov property described earlier in section 2, the process {x k } k≥ k can be regarded as the process which is produced by the numerical method applied to the SDE (2.1) on t ≥ kΔt with the initial data y( kΔt) = xk.On the other hand, let ȳ(t) on t ≥ kΔt be the unique solution of the SDE (2.1) with the initial data ȳ( kΔt) = xk.By the Markov properties of the numerical solution and the true solution as well as the time- k − ȳ(t)| p ≤ C T +1 E|xk| p h(Δt).Moreover, by (2.9) (more precisely, by its equivalent form (2.10)), we have(2.25)E|ȳ(t)|p ≤ M E|xk| p e −λ(t− kΔt) ∀t ≥ kΔt.Using (2.24) and (2.25), we can show, in the same way as we did in Step 1, that (2.26) E|x 2 k | p ≤ E|xk| p e − 1 2 λ kΔt and (2.27) Downloaded 03/20/20 to 80.6.87.97.Redistribution subject to CCBY licenseThe next lemma gives a positive answer to question (Q2) from section 1. Lemma 2.7.Assume that Assumptions 2.4 and 2.5 hold.Assume also that for a step size Δt > 0, the numerical method is pth moment exponentially stable with rate constant γ and growth constant N .If Δt satisfies where T = kΔt and k is the smallest integer which is no less than 4 log(2 p N )/(γΔt), then the SDE (2.1) is pth moment exponentially stable with rate constant λ = 1 2 γ and growth constant M = H(T, p, K)e 1 2 γT , where H(T, p, K) is given in (2.4).Proof.It is easy to see from 4 log(2 p N )/(γΔt) ≤ k that 2 p N e −γ kΔt ≤ e − 3 4 γ kΔt , 1 2 λ(T +1) .The proof is hence complete.c 2015 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license p N e −γT ≤ e − 3 4 γT as T = kΔt.By the elementary inequality (2.18) (2.32) ).
k | p ≤ M E|y(T )| p e −λ(k− k)Δt ∀k ≥ k.c 2015 SIAM.Published by SIAM under the terms of the Creative Commons 4.0 license Downloaded 03/20/20 to 80.6.87.97.Redistribution subject to CCBY license 4.1.Let Assumption 2.1 hold and let p ∈ (0, 1).Assume that the SDE (2.1) is pth moment exponentially stable and satisfies (2.9).Then the solution of the SDE (2.1) satisfies ∈ R n .That is, the SDE is also almost surely exponentially stable.The following theorem is an analogue for the numerical solutions.Theorem 4.2.Assume that the numerical method is pth moment exponentially stable and satisfies(2.11).Then the method satisfies [2] with the initial value y(0) = y 0 ∈ R n , where A i ∈ R n×n ∀0 ≤ i ≤ m.Let us cite a theorem from Arnold, Kliemann, and Oeljeklaus[2].Theorem 4.3.The linear SDE (4.3) is almost surely exponentially stable if and only if it is pth moment exponentially stable for a sufficiently small p ∈ (0, 1).