Issue 
Wuhan Univ. J. Nat. Sci.
Volume 28, Number 5, October 2023



Page(s)  399  410  
DOI  https://doi.org/10.1051/wujns/2023285399  
Published online  10 November 2023 
Mathematics
CLC number: O241.8
Convergence Rates for the Truncated EulerMaruyama Method for Nonlinear Stochastic Differential Equations
^{1}
School of Statistics and Mathematics, Hubei University of Economics, Wuhan 430205, Hubei, China
^{2}
Hubei Center for Data and Analysis, Hubei University of Economics, Wuhan 430205, Hubei, China
^{3}
School of Science, Wuhan University of Technology, Wuhan 430074, Hubei, China
Received:
16
April
2023
In this paper, our main aim is to investigate the strong convergence rate of the truncated EulerMaruyama 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], respectively. In both situations, the common onesided Lipschitz and polynomial growth conditions for the drift coefficients are not required. Two examples are provided to illustrate the theory.
Key words: truncated EulerMaruyama method / strong convergence / moment boundedness
Biography: MENG Xuejing, female, Associate professor, research direction: stochastic differential equations and applications. Email:mengxuejing18@163.com
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 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 superlinearly growing coefficients, implicit EulerMaruyama (EM) methods^{[13]} 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 EulerMaruyama scheme (TEM for short) with strong convergence theory, which stimulates many researchers' interest. There are extensive literatures with the TEM^{[1014]}. These research results are important contributions to numerical approximation theory of SDEs. However, we find that the results normally require the drift and diffusion coefficients satisfy the onesided linear growth condition:
the onesided Lipschitz condition:
and the polynomial growth condition:
The onesided 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 satisfies the onesided 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 onesided polynomially growing whereas the diffusion coefficient is linearly growing, polynomial growing or Hölder continuous. For SDEs with onesided polynomial growing drift and diffusion coefficients, the strong convergence rate is one half.
The next section introduces basic notations and the truncated EulerMaruyama method. After that the strong convergence rate for SDEs with superlinearly growing drift and diffusion coefficients at time was established. Section 2 proves the pathdependent strong convergence rate for SDEs with superlinearly growing drift and linearly growing diffusion coefficients over a finite time interval .
1 SDEs with Polynomial Growing Coefficients
Throughout this paper, unless otherwise specified, let be the Euclidean norm in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by , while its operator norm is denoted by . Let be a complete probability space with a filtration , satisfying the usual conditions (i.e., it is increasing and right continuous and contains all null sets). Let and . Let be the set of all natural integers.
Consider an dimensional stochastic differential system
on with initial data , and are Borelmeasurable.
Assumption 1 (local Lipschitz condition) For each real number , there is a positive constant such that
for with .
Assumption 2 (onesided polynomial growth condition) There exist , and positive constants such that
for .
In this paper, we only require that the drift coefficient satisfies Assumptions 1 and 2. We remove the onesided 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 and (see Ref. [17]). Let be an arbitrary number and be the solution of Eq. (4), define
It is easy to show that there exists a positive constant such that
where 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 of a nonnegative real argument by where are nonnegative real numbers satisfying and . If , then where the nonnegative constant is
We shall now introduce the discrete truncated EM scheme. Choose a strictly increasing continuous function such that as and
Denote by the inverse function of , and . Choose a number , a strictly decreasing function such that
Let the step size be a fraction of , namely for some integer . Define a mapping by , where when . Define the truncated functions
It is easy to see that .
Obviously, the truncated functions and are bounded, although and may not be.
Denote . The discretetime truncated EM numerical solution is defined by
where . The increments are independent distributed Gaussian random variables measurable at the mesh points . Define two continuoustime truncated EM solutions as
where is the indicator function. Clearly, for all .
The truncated functions preserve the Khasminskiitype condition nicely^{[18]}. They cannot preserve Assumption 2 exactly but piecewisely, as described in the following lemma.
Lemma 2 Let Assumption 2 hold. Denote, then for everyand any, wehave
Proof Since is increasing, is decreasing and , we have and for . For any with , it is clear that and hence
For any with , we have and hence
By Assumption 2 and the inequality , we may compute
This completes the proof.
Lemma 3 Fix. Let Assumptions 1 and 2 hold. Then for anyand, there exists , a genericpositive constant dependent onbut independent of, such that
Proof For , there exists a unique nonnegative integer such that . Since , we have
by Eqs. (12) and (13) and the Hölder inequality. For , the Lyapunov inequality gives
This proof is completed.
Let be an arbitrary number and be the continuoustime truncated EM solution defined by Eq. (13), define
Lemma 4 Let Assumption 2 hold, and. Then forand, there exists a positive constantsuch that
Moreover,
Proof We prove the results are true for first. For any , there exists a unique nonnegative integer such that . For , we see that by definition of ; for , we deduce that as well due to for . The Itô formula then gives
Depending on whether , the rest of the proof falls into two cases:
Case 1: For any with , by Lemma 2, we have
According to Lemma and , there is a constant such that
This, together with the Young's inequality for , implies
By the Lyapunov inequality and Lemma 3, the above estimate becomes
Noticing that ,
Using the Gronwall inequality , let , the Fatou lemma gives
Case 2: For any with , we obtain by Lemma 2 :
Note that , then
Recalling that and , by Lemma 1, there is a constant such that
Therefore,
Noticing that , we obtain
The Gronwall inequality and Fatou lemma imply
For , the Lyapunov inequality gives the desired result. The second part of this lemma easily follows.
Lemma 5 Let Assumptions 1 and 2 hold,and. Then for any real numberand, there exists a positive constantsuch that
where .
Proof Denote . Assume first. For given and any real number implies that and
The Itô formula and Assumption 1 give
The Young inequality implies
Applying the Gronwall inequality to the above inequality, we achieve the desired result.
For , picking a , we have
by the Lyapunov inequality.
Theorem 1 Let Assumptions 1 and 2 hold withand, with, and. Then there exists a positive constant such that
Proof For , by the Young's inequality , we have
The Hölder inequality and Lemma 5 imply that
By Lemma 4, we have
This, together with Lemma 5, yields
Choose , then
Example 1Consider a scalar
Denote . Then
Now, we design functions and choose so that condition (10) holds. First, it is easy to see that
Pick , then . Let . Choose satisfying , then . It is easy to see that Assumption 2 holds, by Theorem 1, for any .
In Fig. 1, we plot the truncated EM approximation (11) of Eq. (19) with for initial value . The figure illustrates that the numerical solution has convergence property.
Fig. 1 Numerical simulation of the path with for Eq. (19) 
2 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 . In this section, we consider the pathdependent strong convergence rate over a finite time interval , which requires a stronger assumption on the diffusion coefficient.
Assumption 3 (linear growth condition) For , there is a positive constant such that
Since is linearly growing, it is not necessary to truncate it in this section. Consequently, shall be replaced by in Equations (11) to (13), i.e., in the definition of the discretetime and continuoustime truncated EM solutions.
Lemma 6 Let Assumptions 2 and 3 hold withand. Then for anyandgiven, there exists a positive constantsuch that .
Proof The Itô formula gives
By Lemma 1 , for any and and , there exists a constant such that
Inequality (21) then becomes
It follows that
By the BurkhölderDavisGundy 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 and any with , there exists a positive constant such that
Proof We assume first. For any , there exists a unique nonnegative integer such that .
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 , 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 here may be unbounded because we do not truncate . Thus, the stopping time is necessary to apply the Doob martingale inequality.
Lemma 8 Let Assumptions 2 and 3 hold with and. Then for any, there exists a positive constantsuch that
Proof Let us first assume . Repeating the same process as in the proof of Lemma 4, we obtain . Therefore,
By the BurkhölderDavisGundy inequality, the Hölder inequality and Lemma 7, we may compute
Observing that , we arrive at
The Gronwall inequality yields .
Let , the Fatou lemma gives the desired result.
For , the Lyapunov inequality shall ensure the result.
Lemma 9 Fix and . Let Assumptions 1 to 3 hold with and . Then for and sufficiently small such that , there exists a positive constant such that
Proof Denote . For , we have , and . The Itô formula gives
Here and are the three integrals inside the expression. For , we apply the Hölder's inequality to get
This, together with Assumption 1, implies
For , we have
By the BurkhölderDavisGundy and Hölder inequalities, we may compute
Summarizing up, we reach
This, together with Lemma 7, implies
The Gronwall inequality gives
The proof is completed.
Theorem 2 Let Assumptions 1 to 3 hold withand. Then forand given , the truncated EM scheme described by Equation (13) has the property
Proof Denote . For , the Young's inequality , , gives that
The Hölder inequality and Lemma 9 imply that
By Lemmas 6 and 8, we obtain
Applying Lemma 4, we get
Choosing , we achieve
The proof is completed.
Section 1 discusses the strong convergence rate at time 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 after the th moment uniform boundedness is established.
Example 2 Consider a onedimensional stochastic differential equation
It is easy to see that satisfy Assumptions 1, 2 and 3. Clearly, for . Define , then . Define , obviously, . Choose , then . By Theorem 2, for any .
In Fig. 2, we demonstrate that the numerical solution calculated by Eq. (26) converges strongly in the sense to the numerical exact solution with an order approximately equal to 0.5. We perform 2 000 sample paths and average over all the paths.
Fig. 2 Convergence rate plot for Eq.(26) 
3 Conclusion
In this paper, we discussed the strong convergence of the truncated EM methods for stochastic differential equations with superlinearly growing coefficients. The strong convergence rate at a single time under a very general polynomial growth condition (6) is established in Theorem 1. Imposing the linearly growing condition (20) on the diffusion coefficient, the strong convergence rate over is established in Theorem 2. By using the Gronwall inequality, we proved the th moment uniform boundedness of both the exact and the approximate truncated EM solutions, which is the crucial result in examining the convergence rates.
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All Figures
Fig. 1 Numerical simulation of the path with for Eq. (19) 

In the text 
Fig. 2 Convergence rate plot for Eq.(26) 

In the text 
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