Explicit numerical approximations for stochastic differential equations in finite and infinite horizons : truncation methods , convergence in p th moment and stability

Solving stochastic differential equations (SDEs) numerically, explicit Euler–Maruyama (EM) schemes are used most frequently under global Lipschitz conditions for both drift and diffusion coefficients. In contrast, without imposing the global Lipschitz conditions, implicit schemes are often used for SDEs but require additional computational effort; along another line, tamed EM schemes and truncated EM schemes have been developed recently. Taking advantages of being explicit and easily implementable, truncated EM schemes are proposed in this paper. Convergence of the numerical algorithms is studied, and pth moment boundedness is obtained. Furthermore, asymptotic properties of the numerical solutions such as the exponential stability in pth moment and stability in distribution are examined. Several examples are given to illustrate our findings.


Introduction
In this paper, we study numerical solutions of d-dimensional stochastic differential equations (SDEs) of the form where B(t)isanm-dimensional Brownian motion and f : R d → R d , g : R d → R d×m , which satisfy a local Lipschitz condition, namely, for any N > 0 there is a constant C N such that converge to the true solution of the underlying SDEs. In addition to obtaining the asymptotic pth moment convergence and moment boundedness we consider the approximations to the invariant distributions in infinite horizon. Explicit Euler-Maruyama (EM) schemes are most popular for approximating the solutions of SDEs under global Lipschitz continuously; see, for example, Kloeden & Platen (1992) and Higham et al. (2002). However, many important SDE models satisfy only local Lipschitz conditions or have growth rates faster than linear. For such SDEs, the classical strong convergence for classical EM methods does not hold. Hutzenthaler et al. (2011) showed that the pth moments of the EM approximation for a large class of SDEs with coefficients satisfying super-linear growth diverge to infinity for all p ∈ [1, ∞). Implicit methods were developed to approximate the solutions of these SDEs. Higham et al. (2002) showed that the backward EM schemes converge if the diffusion coefficients are globally Lipschitz while the drift coefficient satisfies a one-sided Lipschitz condition. More details on the implicit methods can be found in Kloeden & Platen (1992), Saito & Mitsui (1993), Hu (1996), Milstein et al. (1998), Burrage & Tian (2002), Appleby et al. (2010) and Szpruch et al. (2011). However, additional computational effort is required for the implementation of the implicit methods.
Since explicit numerical methods have advantages, a couple of modified EM methods have recently been developed for nonlinear SDEs. Hutzenthaler et al. (2012) proposed tamed EM schemes to approximate SDEs with the global Lipschitz diffusion coefficient and one-sided Lipschitz drift coefficient. Sabanis (2013Sabanis ( , 2016 developed tamed EM schemes for SDEs with nonlinear growth coefficients. Moreover, stopped EM schemes (Liu & Mao, 2013), truncated EM schemes (Mao, 2015), multilevel EM schemes (Anderson et al., 2016) and their variants have also been developed to deal with the strong convergence problem for nonlinear SDEs. However, to the best of our knowledge, these modified EM methods still cannot handle the convergence of a large class of SDEs with nonlinear drift and diffusion coefficients, for example, the constant elasticity of volatility model (CEV model) arising in finance for an asset price of the form (Lewis, 2000) dr(t) = β 0 − β 1 r(t) dt + σ r(t) 3/2 dB(t), (1 .3) where β 0 , β 1 , σ are positive constants. Based on the motivation above, we construct easily implementable explicit EM schemes for SDEs with only local Lipschitz drift and diffusion coefficients and establish their convergence. In the process of establishing the strong mean square convergence theory conditionally, Higham et al., (2002Higham et al., ( , p.1060 posed an open problem and noted that 'in general, it is not clear when such moment bounds can be expected to hold for explicit methods with f , g ∈ C 1 .' Despite recent progress in the numerical methods for nonlinear SDEs this problem remains open to date. In this paper, we answer the question of Higham et al. positively by requiring only that the drift and diffusion coefficients are locally Lipschitz and satisfy a structure condition (Assumption 2.1)forthepth moment boundedness of the exact solution for some p ∈ (0, +∞). Talay & Tubaro (1990) investigated the probability law of approximation using the EM scheme for SDE with smooth f and g whose derivatives of any order are bounded. Furthermore, Bally & Talay (1996) expanded the error in power of the step size. Gyöngy (1998) analysed the almost sure convergence. Here we focus on the moment convergence. Higham et al. (2002) and Hutzenthaler et al. (2012) provided the (1/2)-order rate of convergence in moment sense for the backward scheme and the tamed EM scheme under a one-sided Lipschitz condition and polynomial growth for f and global Lipschitz condition for g, respectively. Recently, Sabanis (2016) developed a tamed EM scheme with (1/2)-order rate of convergence. In this paper, we propose a truncation algorithm to relax the restrictions in the studies by Higham et al. (2002) and Hutzenthaler et al. (2012). We demonstrate the convergence Downloaded from https://academic.oup.com/imajna/advance-article-abstract/doi/10.1093/imanum/dry015/4964837 by University of Strathclyde user on 17 April 2018 3 of the algorithm under weaker conditions compared with what is known in the literature. Then under slightly stronger conditions similar to the study by Sabanis (2016) we prove the convergence rate is optimal for the explicit schemes.
While asymptotic properties of the numerical solutions attract more and more attentions (see the studies by Roberts & Tweedie, 1996, Mattingly et al., 2002, Higham et al., 2003and Zong et al., 2016 the moment boundedness of the numerical solutions is also often desirable because its connection to the tightness and ergodicity. However, the classical EM method fails to preserve the asymptotic boundedness for many nonlinear SDEs. For example, Higham et al. (2003) showed that for the nonlinear scalar SDE (1.4) the second moment of the classical EM numerical solution diverges to infinity in an infinite time interval for any given step size and an initial value dependent on the step size. In this paper, as their counterparts of analytic solutions, we show that our explicit schemes will preserve the asymptotic moment boundedness as well as asymptotic stability for a large class of nonlinear SDEs including (1.3) and (1.4) under Assumptions 5.1, 6.1, 7.1. Furthermore, we consider asymptotic properties of our numerical algorithms and demonstrate exponential stability and stability in distribution.
In this paper, adopting the truncation idea from the study by Mao (2015) and using a novel approximation technique, we construct several explicit schemes under certain assumptions on the coefficients of the SDEs and derive convergence results in both finite and infinite time intervals. The numerical solutions at the grid points are modified before each iteration according to the growth rates of the drift and diffusion coefficients such that the numerical solutions will preserve the properties of the exact solution nicely. We approximate the exact solution by piecewise constant interpolation directly, which is different from that of the studies by Higham et al. (2002), Hutzenthaler et al. (2012), Sabanis (2013), Mao (2015) and Bao et al. (2016). Our main contributions are as follows: • An easily implementable scheme is proposed such that its numerical solutions converge to the exact solution in a finite time interval. The rate of convergence is also studied under slightly stronger conditions.
• The open question posed in the study by Higham et al. (2002Higham et al. ( , p.1060) is answered positively. The pth moment of our explicit numerical solution is bounded for the SDEs with only local Lipschitz drift and diffusion coefficients.
• Appropriate truncation techniques and approximation techniques are utilized such that properties of the exact solution are preserved.
• The numerical solutions preserve the pth moment boundedness property of the exact solution almost completely, not only in a finite time interval but also in an infinite time interval for some p > 0.
• Different schemes are constructed to approximate different stochastic dynamical systems that are exponentially stable and/or stable in distribution.
The rest of the paper is organized as follows. Section 2 gives some preliminary results on certain properties of the exact solutions. Section 3 begins to construct an explicit scheme and demonstrate convergence in a finite time interval. Section 4 provides the rate of convergence. Section 5 goes further to obtain the pth moment boundedness in an infinite time interval for some p > 0. Section 6 reconstructs an explicit scheme to approximate the exponential stability. Section 7 analyses the stability of the SDE (1.1) in distribution yielding an invariant measure μ(·). Then another explicit scheme is constructed preserving the stability in distribution and a numerical invariant measure, which tends to μ(·) as the step size tends to 0. Section 8 presents a couple of examples to illustrate our results. Section 9 gives further remarks to conclude the paper.

Preliminaries
Throughout this paper, let Ω, F, F t t≥0 , P be a complete filtered probability space with F t t≥0 satisfying the usual conditions (that is, it is right continuous and F 0 contains all P-null sets). Let T be an m-dimensional Brownian motion defined on the probability space. Let |·| denote both the Euclidean norm in R d and the Frobenius norm in R d×m .AlsoletC denote a generic positive constant whose value may change in different appearances. Moreover, let C 2,1 R d × R + ; R + denote the family of all non-negative functions V(x, t)o nR d × R + , which are continuously twice differentiable in x and once differentiable in t.
For the regularity and pth moment boundedness of the exact solution we make the following assumption.
Assumption 2.1 There exists a pair of positive constants p and λ such that (2.1) Remark 2.2 We highlight that the family of drift and diffusion functions satisfying Assumption 2.1 is large. Denote by C a positive constant. (a) If there are positive constants a, ε and λ such that x T g(x) 2 ≤ a|x| 4−ε + C and that 2x T f (x) +|g(x)| 2 ≤ λ|x| 2 + C then Assumption 2.1 holds for any p > 0.
(d) If there are positive constants a, λ and u > v + 2 such that x T g(x) 2 ≥ λ|x| u + C and that 2x T f (x) +|g(x)| 2 ≤ a|x| v + C then Assumption 2.1 holds for 0 < p < 2.
Proof. It follows from (2.1) that Then for any 0 <κ≪ p|λ|/2, there exists a constant M > 0 such that By the continuity of the functions f and g, It follows from the definition of operator L that Taking ε = κp 2C(p−4) we have Thus, for any p > 0, (2.5) Therefore, it follows from (2.4) and (2.5) that (2.6) The above inequality and Assumption 2.1 guarantee the existence of the unique regular solution x(t) (see the so-called Khasminskii test in the study by Mao & Rassias, 2005). Using Itô's formula, for any 0 ≤ t ≤ T, By Gronwall's inequality we have which implies the desired inequality (2.2).
Remark 2.4 Assumption 2.1 guarantees the existence of global solutions, their regularity and their pth moment boundedness. This is an alternative to Khasminskii's condition that there exist positive constants α, β such that LV p ≤ αV p + β with V = 1 +|x| 2 1/2 . Different from the stability analysis, working with numerical schemes, it is more preferable to use verifiable conditions. As a result, it is more feasible to put conditions on the coefficients of the equations rather than to use an auxiliary function.
Lemma 2.5 Let Assumption 2.1 hold. For each positive integer N > |x 0 | define Then for any T > 0, where C is a generic positive constant dependent on T, p and x 0 and independent of N. Proof. By virtue of Dynkin's formula it follows from (2.6) that

Explicit scheme and convergence in pth moment
In this section our aim is to construct an easily implementable numerical method and establish its strong convergence theory under Assumption 2.1. To define the appropriate numerical scheme we first estimate t h eg r o w t hr a t eo ff and g. Choose a strictly increasing continuous function ϕ : Denote by ϕ −1 the inverse function of ϕ; obviously ϕ −1 :[ ϕ(0), ∞) → R + is a strictly increasing continuous function. We also choose a number △ * ∈ (0, 1) and a strictly decreasing h : (0, where K is a positive constant independent of △. For a given △∈ 0, △ * let us define the truncation mapping π △ : R d → R d by where we use the convention x |x| = 0 when x = 0. Clearly, Next we propose our numerical method to approximate the exact solution of the SDE (1.1). For any given step size △∈(0, △ * ] define where t k = k△, △B k = B t k+1 − B t k . We refer to the numerical method as a truncated EM scheme. The numerical solutions y k are obtained by truncating the intermediate termsỹ k according to the growth rate of the drift and diffusion coefficients to avoid their possible large excursions due to the nonlinearities of the coefficients and the Brownian motion increments. Consequently, we have the following nice linear property Moreover, the truncated EM method is an explicit one so it is easy to use. To proceed, we defineỹ(t) and y(t)byỹ Lemma 3.1 Under Assumption 2.1, the truncation scheme defined by (3.5) has the property Proof. For any integer k ≥ 0wehave Thanks to the Taylor formula, applying the recursion with u > −1, we have where P i (u) represents an ith-order polynomial of u with coefficients depending only on p, and i is an integer. We will prove the result when 0 < p ≤ 2 only; the other cases can be done similarly. It follows from (3.10) that The fact that △B k is independent of F t k implies that This together with (3.2) and (3.6) implies (3.14) Using Also we can prove that, for any i > 3, E ξ i k |F t k = O(△). Combining (3.13)-(3.17) and using (2.1)in Assumption 2.1, for any k ≥ 0, Thanks to the truncated EM scheme (3.5), for any integer k satisfying 0 ≤ k△≤ T, we obtain Solving the above linear first-order difference inequality, we obtain Therefore, we get the desired result that The proof is complete.
Lemma 3.2 Let Assumption 2.1 hold. For any △∈(0, △ * ] define Then for any T > 0, where C is a positive constant independent of △. Proof. We write ρ △ = ρ for simplicity. Then ρ =△β △ , where β △ =:inf k ≥ 0:|ỹ k |≥ϕ −1 (h(△)) . Clearly, ρ and β △ are F t and F t k stopping times, respectively. For ω ∈ β △ ≥ k + 1 we have ỹ k <ϕ −1 (h(△)) and y k =ỹ k , whence it follows from (3.5) that On the other hand, for ω ∈ β △ < k + 1 ,wehaveβ △ ≤ k and hencẽ In other words, we always havẽ where As in the proof of Lemma 3.1 we prove the assertion only for the case when 0 < p ≤ 2; when p > 2it can be done in the same way. Using the technique in the proof of Lemma 3.1 we can show that This together with (3.2) and (3.6) implies We can also prove that for any Combining (3.25)-(3.27), using (2.1) in Assumption 2.1, for any k ≥ 0, (3.29) Solving the above first-order linear inequality leads to Therefore, the desired assertion follows from The proof is complete.
The following theorem presents the pth moment convergence of the truncated numerical solutions.
Theorem 3.3 Under Assumption 2.1, for any q ∈ (0, p), Proof. Let τ N and ζ △ be the same as before. Define θ N, .U s i n g Young's inequality, for any δ>0, we have (3.31) It follows from the results of Theorem 2.3 and Lemma 3.1 that Now let ε>0 be arbitrary. Choose δ>0 sufficiently small for Cqδ/p ≤ ε/3; then we have It follows from the results of Lemmas 2.5 and 3.2 that for any △∈(0, △ * ], Combining (3.31), (3.32) and (3.34), we know that for the chosen N and all △∈(0, △ * ], the desired assertion follows. For this purpose, we define the truncation functions with the initial value z(0) = x 0 .B y( 1.2) in Assumption 2.1, f N (·) and g N (·) are globally Lipschitz continuous with the Lipschitz constant C N . Therefore, SDE (3.36) has a unique regular solution y(t)on t ≥ 0 satisfying On the other hand, for each △∈(0, △ * ], we apply the EM method to SDE (3.36) and we denote by u(t) the piecewise constant EM solution (see Kloeden & Platen, 1992;Higham et al., 2002) that has the property Therefore, (3.35) holds and the desired assertion follows.

Convergence rate
In this section, our aim is to establish a rate of convergence result under Assumption 2.1 and additional conditions on f and g. The rate is optimal, similar to the standard results for the explicit EM scheme with globally Lipschtiz f and g. The work of Higham et al. (2002) gives the optimal rate in qth moment for the implicit EM scheme for q ≥ 2 with global Lipschitz g and a one-sided Lipschitz f together with polynomial growth. Using a similar condition to the study by Higham et al. (2002), the rate for the tamed Euler was obtained (Hutzenthaler et al., 2012).TheworkofSabanis (2016) developed the tamed EM scheme, then obtained the convergence rate under a condition similar to ours. To obtain the rates of convergence we need somewhat stronger conditions compared with the convergence alone, which are stated as follows.
Making use of scheme (3.5) we define an auxiliary approximation process bȳ Note thatȳ t k = y t k = y k , that is,ȳ(t) and y(t) coincide at the grid points.
Lemma 4.4 If Assumptions 2.1 and 4.1 hold with 2(l + 1) ≤ p, for any q 0 ∈ [2, p/(l + 1)], for the process given by (4.6), where C is a positive constant independent of △. Proof. For any t ∈ [0, T] there is a non-negative integer k such that t ∈ t k , t k+1 . Then Due to (4.4), (4.5) and Lemma 3.1, The required assertion follows.
Using techniques in the proofs of Lemmas 3.1 and 3.2, we obtain the following lemmas. Then for any T > 0, where C is a positive constant independent of △.
Theorem 4.8 If Assumptions 2.1 and 4.1 hold with 4(l+1) ≤ p then, for any q ∈ [2, p 0 )∩[2, p/2(l+1)], for the numerical solution defined by (3.5) with ̺ = lq/2(p − q), Remark 4.9 Higham et al. (2002) and Hutzenthaler et al. (2012) obtained the optimal rate 1/2 for the backward EM scheme and the tamed EM scheme of strong convergence under the following condition: the functions f and g are C 1 , and there exists a constant c such that Note that the above condition implies that Assumptions 2.1 and 4.1 hold for any p > 2 and any p 0 > 2.
Thus, under such a condition, in view of Theorem 4.8, the convergence rate of our truncated scheme is optimal. Note that a similar convergence rate result was also obtained by Sabanis (2016) for a modified tamed EM scheme under conditions similar to ours.

The pth moment boundedness in infinite time intervals
Since the moment boundedness in an infinite time interval is related closely to the tightness of the numerical solution, as well as the ergodicity, we go further to realize this property by our explicit numerical solution. Mattingly et al. (2002) showed that for a class of nonlinear SDEs the mean square of the EM numerical solutions in the infinite interval tends to infinity but the mean square of the exact solutions is bounded. Thus, they had to approximate the SDEs by the implicit scheme. Now approximating the exact solutions in an infinite time interval by our numerical method will demonstrate its advantages. First, we give the moment boundedness result on the exact solutions. For convenience, we impose the following hypothesis.
Remark 5.3 Although Assumption 2.1 holds directly from Assumption 5.1 we highlight that the family of the drift and diffusion functions satisfying Assumption 5.1 is large. We give the following examples as special cases in which Assumption 5.1 holds.
Remark 5.4 Assumption 5.1 guarantees the asymptotically pth moment boundedness of exact solutions, which is also an alternative to Khasminskii's condition, which states that there exist positive constants α and β such that LV p ≤− αV p + β with V = 1 +|x| 2 1/2 . Again, for numerical schemes, it is preferable to put conditions on the coefficients as mentioned before.

Exponential stability in pth moment
In this section, we focus on the exponential stability in pth moment. First, we give a sufficient condition for exponential stability in pth moment of the exact solution. Since stability describes the dynamical behavior more precisely than the boundedness, we will construct a truncation mapping and an explicit scheme according to the super-linear growth of the diffusion and drift coefficients. This scheme is suitable for the realization of stability for the nonlinear SDEs. For convenience we impose the following hypothesis.
Assumption 6.1 There exists a pair of positive constants p and λ such that Theorem 6.2 Under Assumption 6.1, the solution x(t) of the SDE (1.1) satisfies where p and λ are given in Assumption 6.1. That is, the trivial solution of the SDE (1.1) is exponentially stable in pth moment.
Proof. It follows from the definition of operator L and Assumption 6.1 that L e pλ 2 t |x| p = e pλ 2 t |x| p pλ 2 Thus, the desired assertion follows from Itô's formula.
Remark 6.3 Assumption 6.1 guarantees the exponential stability of the exact solutions in pth moment, which is also an alternative to Khasminskii's condition, which states that there exists a positive constant α such that LV p ≤− αV p with V = |x|. We use Assumption 6.1 because it is on the coefficients of the SDEs. Note that Assumption 5.1 is sufficient for the boundedness of the pth moment of the analytic solutions but not enough to force the solutions to tend to 0. Thus, for the desired stability, Assumption 6.1 is needed.
Given a step size △∈ 0, △ * , applying the truncation mapping to the truncated EM method yields the scheme To obtain the continuous-time approximation we define u(t)byu(t) := u k for all t ∈ t k , t k+1 . The truncation mapping π 1 △ (x) satisfies (3.4). Thus, Lemma 3.1 and Theorems 3.3 and 5.5 hold for the numerical solution u(t) of the scheme (6.7) under Assumption 6.1. Moreover, π 1 △ (x) has the more precise property (6.6), which may result in the corresponding scheme realizing the exponential stability of the SDE (1.1).

Stability in distribution
This section focuses on asymptotic stability in distribution of SDE (1.1) and the numerical approximation to the invariant measures. In past decades much effort has been devoted to approximating invariant measures for ergodic stochastic processes. Talay (2002) obtained convergence rates for approximation to the invariant measures using an EM implicit scheme for a stochastic Hamiltonian dissipative system with nonglobal Lipschitz coefficients and additive noise. Lamberton & Pagès (2002 studied recursive stochastic algorithms with decreasing step sizes to approximate the invariant distribution for an Euler scheme under Lyapunov-type assumptions under the provision of the existence of such a Lyapunov function. Liu & Mao (2015) took advantage of the implicit backward EM scheme to approximate the invariant measure for nonlinear SDEs with nonglobal Lipschitz coefficients. Mei & Yin (2015) ascertained convergence rates for approximation to invariant measures using EM schemes with decreasing step sizes for switching diffusions. Approximation using EM schemes to the invariant measures for switching diffusions was also dealt with in the study by Bao et al. (2016).
In this paper, we first give sufficient conditions that guarantee SDE (1.1) is asymptotically stable in distribution. Then we construct a truncation mapping and explicit schemes that can approximate the invariant measure of SDE (1.1) effectively. For convenience we impose the following hypothesis.
Assumption 7.1 There exists a pair of positive constants ρ and ν such that Lemma 7.2 Under Assumption 7.1,SDE(1.1) has the property lim t→∞ E|x(t; u) − x(t; v)| ρ = 0 uniformly in u, v ∈ K, (7.2) for any compact subset K ⊂ R d , where ρ is given in Assumption 7.1 and x(t; x 0 ) denotes the unique global solution of SDE (1.1) with the initial value x 0 ∈ R d .
Proof. It follows from SDE (1.1) that By virtue of the definition of the operator L, Then we have Thus, the desired result follows.
Remark 7.3 Assumption 7.1 guarantees the attractivity of the analytic solutions, which is also an alternative to Khasminskii's condition, which states that there exists a positive constant α such that L(|x − y| p ) ≤− α|x − y| p holds for any x, y ∈ R d . As in the other conditions, we prefer to put the conditions on the coefficients of the SDEs for verification purposes.
Proof. We adopt the idea of Mao & Yuan (2006, Theorem 5.43). The main difference is that we remove the linear growth requirement of the drift and diffusion terms. Since the proof is technical we divide it into three steps.
Step 1: Under Assumptions 5.1 and 7.1,S D E( 1.1) has a unique regular solution with an initial value x 0 denoted by x(t; x 0 ), which is a time-homogeneous Markov process. Let P(t; x 0 , ·) denote the transition probability of the process x(t; x 0 ).L e tP R d denote all probability measures on R d . Then for P 1 , P 2 ∈ P R d define a metric d L as where L = l : R d → R : l(x) − l(y) ≤|x − y| and l(·) ≤ 1 .
Since l is arbitrary the desired inequality (7.8) must hold.
Therefore, the desired result follows.
In order to approximate the invariant measure μ of SDE (1.1) we need to construct a scheme such that for any △∈ 0, △ * the numerical solutions are attractive in ρth moment and have a unique numerical invariant measure. However, the truncation mappings π △ (x) and π 1 △ (x) are not suitable for the attractive numerical solutions. Thus, we construct the truncation mapping π 2 △ (x) according to the local Lipschitz growth of drift and diffusion coefficients. Then making use of the appropriate truncation We give the proof outline for the case 0 < p < 2 and other cases can be prove similarly. Using the properties of the Brownian motion (7.13), (7.15), (7.16) and the elementary inequality we can obtain For any given ε ∈ (0, ρν) choose△∈ 0, △ * sufficiently small such that o △ θ 2 ≤ ε/2. It follows from Assumption 7.1 that, for any △∈ 0,△ , Taking the expectation on both sides, letting δ ↓ 0, by the theorem on monotone convergence, we have (7.20) Step 2: The inequality v| p e −(ρν−ε)t k+1 /2 . The desired inequality (7.19) follows from the definition of the numerical solution w(·).
In order to obtain the Markov property of the scheme we state a lemma.

For any
Lemma 7.8 {w k } is a homogenous Markov process with the k-step transition probabilities P which is the desired Markov property. The homogenous property follows from the truncation scheme (7.18) directly.
Next we give a theorem on the asymptotic stability of the scheme.
Theorem 7.9 If Assumptions 5.1 and 7.1 hold, there is a △ 4 ∈ 0, △ * such that for any △∈ 0, △ 4 , the solutions of the truncated EM method (7.18) are asymptotically stable in distribution and admit a unique invariant measure μ △ ∈ P R d .
Proof. Since the proof is rather technical we divide it into three steps.
Step 2: For any given u ∈ R d , there is a 4 ∈ (0, △ 3 ] such that for any △∈(0, is Cauchy in the space P R d with metric d L , namely, there is a positive constant k 1 such that Now for any l ∈ L and any positive integers k, j, compute By virtue of Theorem 5.5 there exists a positive constant △ 1 such that sup 0<△≤△ 1 sup 0≤k<∞ E|w k | p ≤ C. Then there is a positive constant N > |u| sufficiently large such that for any △∈ 0, △ 1 , On the other hand, let △ 4 =△ 1 ∧△ 3 ,by(7.23), for any given △∈(0, △ 4 ], there is a positive integer k 1 satisfying t k 1 = k 1 △≥T 1 such that (7.29) Substituting (7.28) and (7.29)into(7.27) yields El w u k+j − El w u k <ε ∀ k ≥ k 1 , j > 0.
The desired result follows.

Numerical examples
In this section, we consider a number of examples of nonlinear systems and conduct simulations using our numerical schemes.
(8.2) By virtue of Theorem 4.8, the numerical solution of this scheme approximates the exact solution in the mean square sense with error estimate △. It follows from Theorems 6.5 and 7.10 that given η<0, the pth moment of the numerical solution with p < −2η/σ 2 is asymptotically exponentially stable and its measure tends to the Dirac measure as t →∞.
To test the efficiency of the scheme we carry out numerical experiments by implementing (8.2) using MATLAB. We compare the truncated EM method with the backward EM scheme and the tamed EM scheme (see, e.g., Hutzenthaler et al., 2012) numerically. Consider (8.1) with η =− 3/2, σ = 1, ϑ = 1, x 0 = 10 and T = 1. Figure 1 plots the root mean square approximation error E|x(T) − X(T)| 2 1/2 between the exact solution of (8.1) and the numerical solution by the backward EM scheme, the error E|x(T) − Z(T)| 2 1/2 between the exact solution and that of the tamed EM scheme and the error E|x(T) − u(T)| 2 1/2 between the exact solution and that of the truncated EM scheme, as functions of the runtime when △∈ 2 −12 ,2 −13 ,2 −14 ,2 −15 ,2 −16 ,2 −17 . When △=2 −17 , for 1000 sample points, the runtime of X(T) achieving the accuracy 0.0004598 on our computer with Intel Core 2 duo CPU 2.20 GHz, is about 435.4 seconds while the runtime of Z(T) achieving the accuracy 0.000461 is about 364 seconds. The runtime of u(T) achieving the accuracy 0.0004573 is about 362.5 seconds (see Fig. 1). Thus, the convergence speed of the truncated Euler scheme for SDE (8.1) is similar to that of the tamed EM scheme but is 1.2 times faster than that of the implicit backward EM scheme for achieving the same accuracy. Figure 2 gives sample paths of the classical EM solution Y(t) and of the truncated EM solution u(t).
Example 8.2 Because the assumption of constant volatility in the Black-Scholes model has its drawbacks, the formulation of stochastic volatility has attracted much recent attention. One of the popular stochastic volatility models is the risk-adjusted formulation given by Lewis (2000, p.83), dr(t) = β 0 − β 1 r(t) dt + σ |r(t)| 3/2 dB(t), (8.3) r(0) = r 0 > 0 where β 0 , β 1 , σ are positive constants. Such a model is known to possess the so-called mean-reverting property, a direct consequence of which is that the underlying stochastic process is positive recurrent, hence has a stationary distribution. Because the equation does not have an analytic solution, there is a little hope that one can get a closed-form solution for the stationary distribution. Our results obtained in this paper pave a way to numerically approximate the stationary distribution. Note that f (r) = β 0 − β 1 r, g(r) = σ |r| 3/2 satisfy the local Lipschitz condition; moreover, Assumption 5.1 with any 0 < p < 1 and Assumption 7.1 with any 0 <ρ<1 hold. By virtue of Theorems 2.3 and 7.4, equation (8.3) with any initial value r 0 > 0 has a unique regular solution r(t),  First, we generate five sample paths of |r(t)| 1/8 and the sample mean of |r(t)| 1/8 for 4000 sample points in different intervals [0, T], where T = 10, T = 50, T = 100, respectively; see Fig. 3. We compute the sample mean of |r(T)| p for 4000 sample points with T = 10 for different step sizes and different values of p; see Table 1. Figure 4 depicts the frequency of r(T) for 4000 sample points with T = 50, which predicts the stationary distribution.

Concluding remarks
This paper developed numerical solutions of SDEs with truncations. We constructed explicit numerical schemes that allowed both drift and diffusion coefficients to be not globally Lipschitz and to grow faster than linearly. We obtained convergence and moment boundedness of the numerical solutions in infinite time intervals under a local Lipschitz condition and structure conditions required by the analytic solutions. By linking the moment boundedness between the analytic solutions and the explicit numerical solutions for a variety of nonlinear SDEs in finite or infinite time intervals, we answered the open  problem posed in the study by Higham et al. (2002Higham et al. ( , p.1060 positively. Under mild conditions, the (1/2)order rate of convergence is also obtained. Using the features of SDEs, we also studied dynamic behavior including exponential stability and stability in distribution of SDE (1.1). Our results are demonstrated through some examples and numerical experiments.