Convergence Rates for the Truncated Euler-Maruyama Method for Nonlinear Stochastic Differential Equations

: In this paper, our main aim is to investigate the strong convergence rate of the truncated Euler-Maruyama approximations for stochastic differential equations with superlinearly growing drift coefficients. When the diffusion coefficient is polynomially growing or linearly growing, the strong convergence rate of arbitrarily close to one half is established at a single time T or over a time interval [0  T ] , re ‐ spectively. In both situations, the common one-sided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.


Introduction
Stochastic differential equations (SDEs) have been utilized to model various phenomena, such as asset price, SIS epidemic and population dynamics.Analytical solutions can rarely be found for nonlinear SDEs, whereas numerical solutions may be helpful.For SDEs with super-linearly growing coefficients, implicit Euler-Maruyama (EM) methods [1][2][3] have been proposed.In general, the implicit schemes need to solve a nonlinear system at each iteration, and hence requires more computational efforts.
On the contrary, the explicit methods own simple algebraic structure, cheap computational costs and acceptable convergence rate.The explicit EM approximate solution to nonlinear SDEs may diverge to infinity in finite time [4] .Therefore some modified EM methods have been proposed to numerically solve nonlinear SDEs, such as the stopped EM method [5] , the tamed EM method [6] and the tamed Milstein method [7] .Especially, Mao [8,9] invented the truncated Euler-Maruyama scheme (TEM for short) with strong convergence theory, which stimulates many researchers interest.There are extensive literatures with the TEM [10][11][12][13][14] .These research results are important contributions to numerical ap-proximation theory of SDEs.However, we find that the results normally require the drift and diffusion coefficients satisfy the one-sided linear growth condition: the one-sided Lipschitz condition: (xy)( f (x)f (y))≤ L 1 (xy) 2 (2) and the polynomial growth condition: The one-sided Lipschitz condition (2) and polynomial growth condition (3) are frequently assumed in order to establish the strong convergence rates of the implicit EM schemes for highly nonlinear SDEs [15,16] .In this paper, we shall remove conditions (2) and (3).We only need that the drift coefficient f satisfies the one-sided polynomial growth condition to guarantee the strong convergence rates of the underlying numerical solutions, which is much less restrictive than (2) and (3).
The main purpose of this paper is to establish new criteria on the strong convergence rates of the truncated approximation when the drift coefficient is one-sided polynomially growing whereas the diffusion coefficient is linearly growing, polynomial growing or Hölder continuous.For SDEs with one-sided polynomial growing drift and diffusion coefficients, the strong convergence rate is one half.
The next section introduces basic notations and the truncated Euler-Maruyama method.After that the strong convergence rate for SDEs with superlinearly growing drift and diffusion coefficients at time T was established.Section 2 proves the path-dependent strong convergence rate for SDEs with super-linearly growing drift and linearly growing diffusion coefficients over a finite time interval [0T].

SDEs with Polynomial Growing Coefficients
Throughout this paper, unless otherwise specified, let |x| be the Euclidean norm in x Î R n ˉ.If A is a vector or matrix, its transpose is denoted by A T .If A is a matrix, its trace norm is denoted by |A| = trace ( ) A T A , while its operator norm is denoted by   A = sup{|Ax|:|x| = 1}.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 increasing and right continuous and F 0 contains all P-null sets).Let on t ≥ 0 with initial data x 0 Î R n ˉ, and f :R n ˉ® R n ˉg :R n ˉ® R n ˉ´m ˉ are Borel-measurable.Assumption 1 (local Lipschitz condition) For each real number R ≥ 1, there is a positive constant k R such that for Assumption 2 (one-sided polynomial growth condition) There exist n Î N p ≥ 2, and positive constants a 0 aαa i α i i = 12  n such that In this paper, we only require that the drift coefficient f satisfies Assumptions 1 and 2. We remove the one-sided Lipschitz condition (2) in Refs.[8,9] and the polynomial growth condition (3) in Ref. [17].The latter is a vital assumption in establishing the strong convergence rate of the implicit EM scheme for highly nonlinear SDEs.
It is easy to prove that there is a unique global solution to Eq. ( 4) under Assumptions 1 and 2 with α ≥ 2α i and a > ∑ i = 1 n a i (see Ref. [17]).Let R > 0 be an arbitrary number and x(t) be the solution of Eq. ( 4), define It is easy to show that there exists a positive constant c p such that where c p represents a generic positive constant, whose value varies with each appearance throughout the paper.
The following result plays a key role in subsequent sections (For proof, please refer to page 425 of Ref. [3]).
Lemma 1 [3] Define a polynomial function φ(x) of a nonnegative real argument x by φ(x) where the nonnegative constant p 0 is We shall now introduce the discrete truncated EM scheme.Choose a strictly increasing continuous function μ: Denote by μ -1 the inverse function of μ, and μ -1 :[μ(0) + ¥)® R + .Choose a number D * Î(01), a strictly decreasing Let the step size D Î ( 0D * ) be a fraction of T, namely D = T/M for some integer M. Define a mapping π: , where x/|x| = 0 when x = 0. Define the truncated functions It is easy to see that Obviously, the truncated functions f D and g D are bounded, although f and g may not be.
Denote t k = kD.The discrete-time truncated EM numerical solution X D ( t k )( » x ( t k )) is defined by where Dw t k = w ( t k + 1 ) -w ( t k ) .The increments Dw t k are independent N(0D)-distributed Gaussian random variables F t k - measurable at the mesh points t k .Define two continuous-time truncated EM solutions as where I [ ) The truncated functions preserve the Khasminskii-type condition nicely [18] .They cannot preserve Assumption 2 exactly but piecewisely, as described in the following lemma.
Lemma 2 Let Assumption 2 hold.Denote a͂ = ( ∑ i = 1 ¥ a i + a 0 ) /2, then for every D Î ( 0D * ) and any x Î R n ˉ, we have By Assumption 2 and the inequality ab ≤ a 2 + b 2 2 , we may compute ) This completes the proof.
Lemma 3 Fix T > 0. Let Assumptions 1 and 2 hold.Then for any D Î ( 0D * ) and p > 0, there exists c p , a generic positive constant dependent on Tpk R but independent of D, such that by Eqs. ( 12) and ( 13) and the Hölder inequality.For 0 < p ˉ< 2, the Lyapunov inequality gives This proof is completed.
Let R > 0 be an arbitrary number and x D (t) be the continuous-time truncated EM solution defined by Eq. ( 13), define there exists a positive constant c p such that sup Moreover, Proof We prove the results are true for p ≥ 2 first.For any s > 0, there exists a unique nonnegative integer k such Depending on whether | x ˉD (s) | ≤ μ -1 (h(D)), the rest of the proof falls into two cases: Case 1: For any x ˉD (s , by Lemma 2, we have ) ) According to Lemma 1α ≥ 2α i and a > ∑ i = 1 n a i , there is a constant c 0 such that This, together with the Youngs inequality By the Lyapunov inequality and Lemma 3, the above estimate becomes Noticing that sup Using the Gronwall inequality , let R ® ¥, the Fatou lemma gives , we obtain by Lemma 2 : Therefore, Noticing that sup The Gronwall inequality and Fatou lemma imply For 0 < p ˉ< 2, the Lyapunov inequality gives the desired result.The second part of this lemma easily follows.
Lemma 5 Let Assumptions 1 and 2 hold, p > 0α ≥ 2α i (i = 1n) and a > ∑ i = 1 n a i .Then for any real number R < μ -1 (h(D)) and D Î ( 0D * ) , there exists a positive constant c p such that Proof Denote e(t) = x(t) -x D (t).Assume p ≥ 2 first.For given D Î ( 0D * ) and any real number The Itô formula and Assumption 1 give Applying the Gronwall inequality to the above inequality, we achieve the desired result.For 0 < p ˉ< 2, picking a p > 2 , we have pq ) p > 2, and q Î[2p).Then there exists a positive constant c p such that Proof For p ≥ 2, by the Youngs inequality x q y ≤ δq p x p + pq pδ q pq y q pq "xyδ > 0, we have The Hölder inequality and Lemma 5 imply that By Lemma 4, we have This, together with Lemma 5, yields Now, we design functions μ(x)h(x) and choose D * so that condition (10) holds.First, it is easy to see that Pick μ(u) = 5u 5 u ≥ 1, then μ -1 (u) = ( u 5 ) In Fig. 1, we plot the truncated EM approximation (11) of Eq. ( 19) with T = 2 for initial value x(0) = 1.The figure illustrates that the numerical solution has convergence property.

Strong Convergence Rate over a Finite Time Interval
Section 1 has established the strong convergence rate of the truncated EM solution at a fixed time T > 0. In this section, we consider the path-dependent strong convergence rate over a finite time interval [0T], which requires a stronger assumption on the diffusion coefficient.
Assumption 3 (linear growth condition) For x Î R n ˉ, there is a positive constant k g such that Since g is linearly growing, it is not necessary to truncate it in this section.Consequently, g D (×) shall be replaced by g (×) in Equations ( 11) to ( 13 By Lemma 1 , for any x Î R n ˉ and α ≥ 2α i (i = 12n) and a > ∑ i = 1 n a i , there exists a constant c 0 such that It follows that By the Burkhölder-Davis-Gundy inequality, we obtain The above estimate, together with Assumption 3, yields Finally, the desired result is obtained by the Gronwall inequality and the Fatou Lemma.
From the procedure of the proof in Lemma 6, we can see that Assumption 3 plays an important role in establishing the estimate (22), which makes the Gronwall inequality applicable to establish the moment boundedness.Lemma 7 Let Assumption 3 hold.Then for p > 0 and any D Î ( 0D * ) with h(D)≥ 1, there exists a positive con- Proof We assume p ≥ 2 first.For any t ≥ 0, there exists a unique nonnegative integer By the linear growth condition (3) and the Doob martingale inequality, it is easy to see that This, together with the Fatou Lemma, implies that For 0 < p ˉ< 2, the Lyapunov inequality gives The proof of the above lemma is different from that of Lemma 3 in that: ) is bounded in Lemma 3, but g here may be unbounded because we do not truncate g.Thus, the stopping time is necessary to apply the Doob martingale inequality.
Lemma 8 Let Assumptions 2 and 3 hold with α ≥ 2α i (i = 12n) and a > ∑ i = 1 n a i .Then for any p > 0, there exists Proof Let us first assume p ≥ 2. Repeating the same process as in the proof of Lemma 4, we obtain By the Burkhölder-Davis-Gundy inequality, the Hölder inequality and Lemma 7, we may compute Observing that E é ë Let R ® ¥, the Fatou lemma gives the desired result.For 0 < p ˉ< 2, the Lyapunov inequality shall ensure the result.
Lemma 9 Fix T > 0 and p ≥ 2. Let Assumptions 1 to 3 hold with α ≥ 2α i (i = 12n) and a > ∑ Here J 1 J 2 and J 3 are the three integrals inside the expression.For J 1 , we apply the Hölders inequality to get This, together with Assumption 1, implies For J 2 , we have By the Burkhölder-Davis-Gundy and Hölder inequalities, we may compute Proof Denote e(t) = x(t) -x D (t).For p > 2, the Youngs inequality x q y ≤ δq p x p + pq pδ q pq y q pq xy, δ > 0, gives that Applying Lemma 4, we get Choosing δ = D q 2 h(D) q R = ( D q 2 h(D) q ) -1 pq , we achieve E é ë ê ê ê ê sup The proof is completed.Section 1 discusses the strong convergence rate at time T under a very general polynomial growth condition (6).Imposing the linear growth condition (20) on the diffusion coefficient, we obtain the strong convergence rate over [0T] after the p-th moment uniform boundedness is established.