Open Access
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

© Wuhan University 2023

Licence Creative CommonsThis 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 super-linearly growing coefficients, implicit Euler-Maruyama (EM) methods[1-3] 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[10-14]. 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 one-sided linear growth condition:

(1)

the one-sided Lipschitz condition:

(2)

and the polynomial growth condition:

(3)

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 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 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 .

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

(4)

on with initial data , and are Borel-measurable.

Assumption 1 (local Lipschitz condition) For each real number , there is a positive constant such that

(5)

for with .

Assumption 2 (one-sided polynomial growth condition) There exist , and positive constants such that

(6)

for .

In this paper, we only require that the drift coefficient 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 and (see Ref. [17]). Let be an arbitrary number and be the solution of Eq. (4), define

(7)

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

(8)

We shall now introduce the discrete truncated EM scheme. Choose a strictly increasing continuous function such that as and

(9)

Denote by the inverse function of , and . Choose a number , a strictly decreasing function such that

(10)

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 discrete-time truncated EM numerical solution is defined by

(11)

where . The increments are independent -distributed Gaussian random variables -measurable at the mesh points . Define two continuous-time truncated EM solutions as

(12)

(13)

where is the indicator function. Clearly, for all .

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, then for everyand any, wehave

(14)

Proof   Since is increasing, is decreasing and , we have and for . For any with , it is clear that and hence

(15)

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 continuous-time truncated EM solution defined by Eq. (13), define

(16)

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

(17)

The Hölder inequality and Lemma 5 imply that

(18)

By Lemma 4, we have

This, together with Lemma 5, yields

Choose , then

Example 1Consider a scalar

(19)

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.

thumbnail 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 path-dependent 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

(20)

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 discrete-time and continuous-time 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

(21)

By Lemma 1 , for any and and , there exists a constant such that

Inequality (21) then becomes

It follows that

By the Burkhölder-Davis-Gundy inequality, we obtain

The above estimate, together with Assumption 3, yields

(22)

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 Burkhölder-Davis-Gundy 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 Burkhölder-Davis-Gundy and Hölder inequalities, we may compute

Summarizing up, we reach

(23)

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

(24)

The Hölder inequality and Lemma 9 imply that

(25)

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 one-dimensional stochastic differential equation

(26)

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.

thumbnail 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 super-linearly 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

thumbnail Fig. 1

Numerical simulation of the path with for Eq. (19)

In the text
thumbnail Fig. 2

Convergence rate plot for Eq.(26)

In the text

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