Abstract
The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSM methods is investigated. Furthermore, the stability regions of the DSSM methods are compared with those of test equation, and it is proved that the methods with are stochastically A-stable. Second, the nonlinear stability of DSSM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.
1. Introduction
Many real-world phenomena in different fields of science, such as biology, financial engineering, neural network, and wireless communications, can be simulated by the Itô stochastic differential equations (SDEs) of the formwhere is the drift coefficient and is the diffusion coefficient and the Wiener process is defined on a given probability space with a filtration which satisfied the usual conditions. In order to describe the properties of SDEs systems, numerical solutions have attracted a lot of attention. There is an extensive literature concerned with explicit and implicit numerical methods for SDEs (see [1–7]).
In order to get insight into the stability behavior of the numerical methods for SDEs (1), the scalar equation has been used as a test problem as follows:where . Mean-square (MS) stability conditions for several numerical methods have been derived (see [8–10]). Saito and Mitsui [10] proposed the concept of the MS stability for a numerical methods solving (2). The MS stability of the classes of theta methods for scalar SDEs (2) is introduced in [11]. Higham [9] introduced allowing in the semi-implicit Milstein scheme benefits in terms of the stability for (2).
In literature, there are several attempts to constructing numerical methods based on split-step techniques to improve the fundamental analysis that contains convergence and stability for SDEs. Various implicit split-step numerical methods have been derived based on Euler method. For example, Higham et al. [12] derived the split-step backward Euler (SSBE) method. In addition, the split-step theta (SS) methods which generalize the SSBE method when were discussed in [13, 14]. Although these methods are considered stochastically A-stable (i.e., the MS stability region of the test equation is subset of or equal to that of numerical method for any step-size (see [15])), these methods have convergence order 0.5.
Recently, there are many split-step methods that have been constructed based on Milstein method, to improve the fundamental analysis that contains convergence and stability for SDEs. For example, in 2009, Wang and Liu [7] presented the drifting split-step backward Milstein (DSSBM) method. Moreover, in 2015, Voss and Abdul Khaliq [6] combined the predictor corrector method with a Milstein method to investigate split-step Adams-Moulton Milstein (SSAMM) method. The two-step Milstein schemes are defined, Adams-Bashforth Milstein (ABM), Adams-Moulton Milstein (AMM), and two-step midpoint-Simpson Milstein (BDFM) schemes by Tocino and Senosiain [15], in 2015. The necessary and sufficient conditions for their MS stability are given, and stability regions and A-stable approaches have been discussed. Although these methods improved the convergence order to be 1.0, unfortunately, we can see that the MS stability conditions of these methods have some restriction on the parameters and step-size. Furthermore, these methods are not A-stable.
There are a few works that discussed the nonlinear stability of numerical methods for SDEs (1). The exponential MS stability of the Euler and SSBE methods has been proved for nonlinear SDEs, under the assumption that the drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient satisfies a global Lipschitz condition (see [16]). Furthermore, without the global Lipschitz condition, the two classes of theta methods were proved to reproduce the exponential MS stability of the exact solution (see [13, 17, 18]). Note that these numerical methods converge with order 0.5. Recently, the exponential MS stability has been discussed for split-step Milstein type methods, which converge with order 1.0. In 2016, Jiang et al. [19] proved that the double-implicit and split two-step Milstein schemes can preserve the exponential MS stability of the original SDEs under Lipschitz condition with restriction on step-size. Nevertheless, the MS stability analysis of the multistep methods is focused on linear SDEs, and there exist very little results on the stability analysis for nonlinear SDEs. So, the particular literature is still limited and welcomes attempts to fill this gap.
Following the MS stability analysis of Milstein type methods for linear and nonlinear SDEs. Zong et al. [20] discussed the exponential MS stability of two classes of theta Milstein methods, the split-step theta Milstein (SSTM), and stochastic theta Milstein (STM) methods. With restriction on parameters and step-size, they proved the MS stability under of two numerical methods. However, they did not discuss the stability regions and A-stability approaches. Also, with restriction on step-size, the authors proved that under one-sided Lipschitz condition the two classes of theta Milstein methods can share the exponential MS stability of the exact solution for nonlinear SDEs.
Nowadays, split-step numerical methods have attracted a lot of attention from scholars for SDEs and have been proven to be a very efficient approach. In order to improve the fundamental analysis of numerical methods, the drifting spit-step theta Milstein (DSSM) methods are considered for solving SDEs (1), in this paper. First, the MS stability results of the methods are investigated for scalar linear SDEs (2) as follows: (a) If , the DSSM methods are MS stable with restriction on step-size. (b) If , the proposed methods are MS stable for all under constraints on the parameters , and . (c) The restrictions on the parameters can be avoided when and the numerical methods are stochastically A-stable. The regions of MS stability are examined to explain the effectiveness of the methods. Second, we prove that the DSSM methods are MS stable for all step-size with under a coupled condition on the drift and diffusion coefficients for nonlinear SDEs (1). Finally, we give a linear and nonlinear numerical examples to check the accuracy of the proposed methods in the light of the stability conditions, especially the value of parameter . This work is different from [20] in that we extend the stability results for asymptomatic MS stability with . In addition, we prove for , the DSSM methods can preserve the asymptomatic MS stability of the exact solution with no restriction on parameters and step-size. Furthermore, we discussed the MS stability regions and A-stable approaches of DSSM methods for linear SDEs. Also, we proved that, under local Lipschitz condition, the numerical methods can preserve the asymptomatic MS stability of the nonlinear SDEs without restriction on step-size.
The rest of this paper is organized as follows. In Section 2, we present some necessary notations and preliminaries, then the drifting split-step theta Milstein methods are given. The linear MS stability of the proposed methods is studied, in Section 3. In Section 4, our attention is turned to the nonlinear stability of numerical methods for SDEs. In Section 5, numerical results are given in order to demonstrate the accuracy of the proposed methods under the stability conditions and compared with existing methods. Conclusion is given in Section 6.
2. Notations and Preliminaries
Let be a complete probability space with a filtration , which satisfies the usual conditions (i.e., the filtration is right-continuous and each , contains all -null sets in ). Let be one-dimensional Brownian motion defined on this probability space and be -adapted and independent of . is the Euclidean norm in . presents and presents . represents the positive integer set, namely, . Moreover, we assume to be -measurable and .
Let be Borel measurable functions. Let us consider the -dimensional SDEs (1) with initial data . Assume that and satisfy the following assumption.
Assumption 1 (local Lipschitz condition). There exists a positive constant for all such that for all and with ,
Assumption 2. Assume that there exists a constant such that for any , the function in (1) also satisfies
From [21], we note that if SDEs satisfy Assumption 1, then the linear growth condition or a class of monotone conditions will ensure a global solution for the SDEs (1).
In order to improve the fundamental analysis that contains convergence and stability of numerical solutions for SDEs (1), there are many drifting split-step methods that have been constructed over the past several years [6, 7, 12–14, 20, 22]. In this paper, we consider general split-step numerical methods; the drifting split-step theta Milstein (DSSM) methods arewhere is approximation to at the discrete points in time , with step-size , is a given positive integer, is a free parameter, with increments being independent -distributed Gaussian random variables and . Moreover, is -measurable at the mesh point . The DSSM ((5)-(6)) with parameter reduce to the DSSBM [7], while with parameter , the DSSM become the classical Milstein [23].
The methods ((5)-(6)) are discussed in [20], where it is called the split-step theta Milstein (SSTM) method. The strong convergence order 1.0 of the DSSM methods ((5)-(6)) for SDEs (1) has been given in [20, 22]. Here, we only list the definition of the numerical solutions and omit the counterparts for the continuous solutions to the SDEs. We will further explain the relevance of these concepts to numerical theory and practice in Section 4.
Definition 3. The numerical method is said to be MS stable, if there exists , such that any application of the method to SDEs (1) generates numerical approximation , which satisfyfor all step-size .
3. Linear Stability
Recently, split-step numerical methods have attracted a lot of attention from scholars SDEs and have been proven to be a very efficient approach. It is well known that there are many split-step numerical methods with convergence order 0.5 that are A-stable for scalar SDEs (2), for example, SSBE and SS methods. In addition, split-step Milstein schemes with convergence order 1.0 have been introduced for SDEs in literature review such as SSAMM, DSSBM, and (ABM, AMM, BDFM) methods [6, 7, 15], respectively. Unfortunately, when comparing the stability regions of these Milstein type schemes with that of test equation (2), we can see that these methods are not A-stable, and the MS stability conditions of these methods have some restriction on parameters and step-size. In this section, we discuss the MS stability of new general split-step numerical methods, the DSSM methods ((5)-(6)) for scalar linear SDEs (2). First, we recall some notations of stability.
Definition 4 (see [10]). Suppose that the equilibrium position of SDEs is asymptotically MS stable. Then a numerical method that produces the iterations to approximate the solution of SDEs (1) is said to be asymptotically MS stable if
We can apply one-step method to the test equation (2) which is represented bywhere is the standard Gaussian random variable . The following condition characterizes the MS stability of (2)
Lemma 5 (see [10]). If the constants in (2) satisfythen the solution of (2) is asymptotically stable in the MS sense; that is,
Definition 6 (see, [10]). The numerical method is said to be MS stable for , , and ifwhere is called MS stability function of the numerical method.
Applying the DSSM methods ((5)-(6)) to the scalar linear SDEs (2), we obtained the following form:
During the rest of this section, following Higham [9], we extend the stability results in [20] for asymptotic MS stability with . In addition, we prove that (a) for , the DSSM methods can preserve the asymptotic MS stability of the exact solution with no restriction on parameters and step-size. Furthermore, the methods are stochastically A-stable. (b) The stability regions of the DSSM methods are compared with those of considerable numerical methods to explain the effectiveness of the proposed methods.
Now, we can state the main results of the linear stability of the DSSM methods ((5)-(6)) for SDEs (2) in the following theorem.
Theorem 7. Supposing that MS stability condition (10) holds, one has the following: (1)If , then the DSSM methods are MS stable for all , (2)If and , then the DSSM methods are MS stable for all .(3)If and , then the DSSM methods are MS stable for all .
Proof. The proof of result number , when , has been given in ([20], Theorem 5.3). So, we omit the proof here. In the following, we prove results numbers and , when .
Note that is -measurable at the mesh point ; we easily know from ((5)-(6)) that is also -measurable at related mesh-points, and is independent of . From , , , and , it is easy to see from (13) thatwhereHence if and only ifWith respect to condition (10), we know that and the following results can be obtained.
If and , that is,then the DSSM methods are MS stable for .
If and , that is,then (17) holds for all . Namely, the DSSM methods are MS stable for all .
The stability regions of the proposed methods with are strictly contained in that for the problem. Theorem 7 shows that this behavior extends to DSSM methods (13) with when the diffusion term dominates. However, Theorem 7 also shows that if the drift term dominates, then the unconditional stability holds.
The following corollary shows that, for , the DSSM methods are MS stable with no restrict on the parameters.
Corollary 8. If , then the DSSM methods (13) are MS stable for all .
Proof. Condition (10) and implywhich yields (17). Hence, the DSSM methods are MS stable for all .
Remark 9. Zong et al. ([20], Theorem 5.3) discussed the exponential MS stability of the DSSM (13) with . The authors proved that, for , the methods can share the exponential MS stability of the exact solution with restriction on step-size. For , if the diffusion term plays a crucial role, then the restriction on the step-size still holds with restriction on parameters, and if the drift term plays a crucial role, then the numerical methods are MS stable for all step-size with restriction on parameters. In this paper, Theorem 7 extended the same results with free parameter , which can be greater than 1. In addition, Corollary 8 proved that for , the DSSM methods can preserve the asymptomatic MS stability of the exact solution for SDEs (2) with no restriction on parameters and step-size.
By substituting into (6), the approximation in the DSSM ((5)-(6)) methods is in fact the stochastic theta Milstein (STM) approximation, which is provided for linear SDEs (2)The exponential MS stability of STM (21) methods was investigated for linear SDEs with restriction on step-size in [20]. Theorem 7 and Corollary 8 illustrated the asymptotic MS stability results of the STM (21) for linear SDEs (2).
The MS stability regions are the areas under the plotted curves and symmetric about the plane with and . In this case, the stability conditions (MS stability) (10) and (17) for the test problem and DSSM methods, respectively, become
We plot the stability regions of the DSSM and test problem as follows. Let denote the MS stability region of the SDEs (2), and let , (23) hold be the MS stability regions of DSSM methods. Figure 1 illustrates how the varies with . The green shading marks the regions and the red shading superimposes for various values of . Figure 1 confirms our results.
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Remark 10. By comparing the MS stability regions of DSSM methods with various values of parameter and that of scalar SDEs (2) (see Figure 1), we can see that the DSSM methods are stochastically A-stable with .
In Figure 2, we compare the MS stability regions of the proposed DSSM methods (23) (on the figure, DSSTM) with various value of parameter with those of test equation (22), Milstein, drifting split-step backward Milstein (DSSBM), and split-step Adams-Moulton Milstein (SSAMM) methods (note that Voss and Abdul Khaliq [6] presented that the SSAMM methods with give the large MS stability regions). The MS stability functions (MS stability conditions) of these numerical methods are shown in Table 1. We examine the MS stability regions of the DSSM methods with , respectively. From Figure 2, we can see that when , the MS stability regions of the DSSM methods and those of the DSSBM method match exactly (see Figure 2(a)). With , the MS stability regions of the DSSM methods are identical exactly to the test problem stability regions. Also, the MS stability regions of DSSM methods are better than those of the others. Furthermore, when , we note that the test problem stability regions are subset of those of the proposed methods; that is, the DSSM methods are stochastically A-stable under with stability regions being better than those of existing Milstein type methods.
(a) ,
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(c) ,
Remark 11. If we look closely to the stability conditions of the split-step methods which are based on Milstein method to approximate the diffusion part in SDEs (2), we can find that the term plays an important role in the stability functions (see Table 1). In addition, this term should be handled to improve the stability properties of the split-step method by the one-step method which is used for the drift part in SDEs (2). We can see from Table 1 that the backward Euler method and the predictor corrector method could not handle this term in DSSBM and SSAMM methods, respectively. So, the MS stability properties of these methods are not A-stable and have restriction on the parameters and step-size. In the case of the proposed DSSM methods, we find that the free parameter plays the main role to handle that term . So, the MS stability properties of the DSSM methods are A-stable with (see Figure 1) and have no restriction on the parameters and step-size (see Corollary 8).
4. Nonlinear Stability
In this section, we discuss the nonlinear stability of the DSSM methods for SDEs (1). It is impossible to find a sufficient and necessary condition for analytical stability for nonlinear SDEs. For the purpose of stability, assume that . This shows that SDEs (1) admit a trivial solution. Then inequality (4) reduces to
We assume that and satisfy Assumption 1 (local Lipschitz condition), which is classical for the nonlinear SDEs. To investigate stability of numerical approximation, let us firstly give the stability criterion of SDEs (1).
Theorem 12 (see [21]). Let Assumption 1 hold. If there exists a positive constant such that for all ,then the solution of (1) is MS stable.
Following Higham [9], we can prove that the DSSM methods share the asymptotic MS stability of the exact solution when with no restriction on step-size. Before giving the main theorem of the MS stability, we establish the key role in the following lemma.
Lemma 13. Let conditions ((24), (25)) hold. Moreover, the functions and satisfy the local Lipschitz condition (3) for any , with a positive constant for all . Then there exists a constant for all and , such that the DSSM methods ((5)-(6)) have the properties
Proof. For sufficiently large , we define the stopping timeIt is observed that, for ,Then we can complete the proof of our theorem in the same idea and similar process, which were used in the first part of the proof of Theorem 5.2 in [20]. So, we omit the detailed proof here.
Theorem 14. Let conditions ((24), (25)) hold. Then the DSSM methods ((5)-(6)) have the following stability results:(1)If and for all , then the DSSM methods are MS stable for all .(2) If and if there exists a positive constant such that, for any , then the DSSM methods are MS stable for all ,
Proof. By (5), we getSquaring both sides of (6) and substituting (32), we havewhereNote that is -measurable at the mesh point , and we easily know from ((5)-(6)) that is also -measurable at related mesh point, and is independent of . So, , , , and .
Case I. If , then . Hence, we need to discuss the sign of the following term:First. If , we obtain from (33), with respect to condition (25) and Lemma 13, that the DSSM methods are MS stable for all .
Second. If , using condition (24), we obtain from (33)Taking expectation and summation on both sides of (36) over from with respect to Lemma 13 givesLetThen for any , the DSSM methods are MS stable.
Case II. If , using conditions ((24), (30)), we get from (33)Letwith respect to Lemma 13. We get that the DSSM methods are MS stable for any .
Remark 15. Zong et al. ([20], Theorem 5.2) proved that, under one-sided Lipschitz condition, when the DSSM methods can share the exponential MS stability of the exact solution with restriction on step-size. In this paper, Theorem 14 proved that, under local Lipschitz condition when , the DSSM methods can preserve the asymptotic MS stability of the exact solution with restriction on step-size. In addition, with suitable condition on the drift and diffusion functions (29), when the numerical methods can preserve the asymptotic MS stability of the exact solution with no restriction on step-size.
From the extension of MS stability for the semi-implicit theta Milstein methods [9] and the results of the DSSM methods ((5)-(6)) for linear SDEs in Section 3, we know that the stability for the methods with is strictly contained in that for the problem. Theorem 14 shows that these behaviors extend to the numerical methods with when the diffusion term dominates. Also, if the drift term dominates, then the unconditional stability holds. The following remark shows that, for , the numerical methods are MS stable.
Remark 16. In Theorem 14, if , then the DSSM methods ((5)-(6)) are MS stable for all , under a coupled condition on the drift and diffusion coefficients (29). In case of , this condition reduces to
From Remark 16, we get that the approximation solution can reproduce the MS stability of trivial solution when . To illustrate these results, we consider the following nonlinear SDEs:
Clearly, the coefficients of (42) satisfy conditions ((3), (4)). Now, we explain that the stability condition (41) of the numerical methods satisfies the exact solution stability condition (25).
The Exact Solution Stability Condition
The Stability Condition for DSSM Methods We get
Remark 17. By substituting into (6), the approximation in the DSSM methods ((5)-(6)) is in fact the stochastic theta Milstein (STM) approximationThe exponential MS stability of STM (46) is investigated for nonlinear SDEs with restriction on step-size in [20]. Theorem 14 and Remark 16 illustrate the asymptotic MS stability results of the STM (46) without restriction on step-size for SDEs (1).
5. Numerical Experiments
In order to examine the accuracy of the proposed methods under the stability conditions, we consider several illustrative numerical examples for showing the strong convergence order and MS stability of DSSM methods for SDEs. The MS errors at time versus the step-size are analyzed under different values of the parameter in a log-log diagram. The MS errors of the numerical approximations are defined by [14] as follows:where is a numerical approximation to and is the value of the exact solution of SDEs at time . The superscript means the sample path, .
Example 18 (scalar linear SDEs). We apply the DSSM methods to the scalar linear SDEs whose exact solution isWe use the parameters and the same as in [6, 14] and demonstrate the strong convergence rate of the DSSM methods at the terminal time . We compute different discretized Brownian paths over with step-size . For each path, the DSSM methods are applied with five different step-sizes: , . Table 2 compares the mean of absolute errors over the sample paths for SS methods [14] and SSAMM methods [6], with the proposed DSSM methods (note that the value of parameter for SS and SSAMM has been chosen to give the best absolute errors of those methods according to [6, 14]). We find that the proposed DSSM methods are more efficient than SS and SSAMM methods. Figure 3 shows the results of the MS errors at time versus the step-size under different values of the parameter in a log-log diagram. This explains that there is a balance between the value of parameter and the accuracy.
Example 19 (scalar nonlinear SDEs). We consider the nonlinear SDEswhere with and being real constants. The exact solution is given by [24] Errors of SS methods [14] and SSAMM methods [6] with our DSSM methods are displayed in Table 3 at the terminal time . The numerical results provide a comparison of these methods for fixed parameters and . We compute 4000 different discretized Brownian paths over with step-size . For each path, the methods are applied with five different step-sizes: , . We can see that the proposed methods return the most accurate solution. Figure 4 shows the results of the mean of absolute errors using a log-log plot. This explains that the absolute error with is the best one of the DSSM methods.
Finally, we can see that the numerical examples show a balance between the stability conditions which depends on values of parameter and the accuracy of the proposed methods in approximation of SDEs. Furthermore, numerical examples show that the numerical methods are still effective with .
In the following, we illustrate the stability properties of the DSSM methods by simulating SDEs (2). The set of coefficients satisfy condition (10). So, the trivial solution of the test equation is MS stable. The data used in Figures 5, 6, and 7 is obtained by the MS of data from 4000 trajectories, that is, .
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We test the MS stability for the DSSM methods with , and the initial value . For and , we obtain and . First, we fix the parameter and and change the step-size (see Figures 5 and 6, respectively). It is shown that the DSSM methods are MS stable for any if . Figure 7 explains that, for any , the DSSM methods are MS stable for any .
6. Conclusion
We are interested in mean-square (MS) stability of the drifting split-step theta Milstein (DSSM) methods for stochastic differential equations (SDEs). Zong et al. [20] proved that, under one-sided Lipschitz condition, the numerical methods with can share the exponential MS stability of the exact solution for nonlinear SDEs with restriction on step-size. Also, the authors could prove that, for , the numerical methods are exponential MS stable for linear SDEs for all step-size with restriction on parameters. In this work, for linear SDEs, we extended the stability results in [20] for asymptotic MS stability with . In addition, we proved that, for , the DSSM methods can preserve the MS stability of the exact solution with no restriction on parameters and step-size. Furthermore, by comparing the stability regions of the numerical methods with those of the test equation, we proved that the DSSM methods are stochastically A-stable with . The stability regions of various Milstein type schemes were compared to show the effectiveness of the proposed methods. For nonlinear SDEs, under local Lipschitz condition, we proved that the DSSM methods with are asymptotic MS stable with no restriction on step-size under suitable condition on the drift and diffusion functions. In addition, we gave an example to explain that, for , the suitable condition (stability condition) of the numerical methods satisfies that of the exact solution. Finally, numerical examples are given to show that there is a balance between the stability conditions which related to values of parameter and the accuracy of the proposed methods in approximation of SDEs. Furthermore, numerical examples show that the numerical methods are still effective with .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful to Dr. Zhanwen Yang for helpful discussions and suggestions. This research was partly financed by NSFC Grants 11771111 and 91646106 and China Aerospace Science and Technology Corporation: JSKFJ201604120014. Additionally, this work is supported by Higher Education Commission of Egypt.