An improved Milstein method for stiff stochastic differential equations
 Zhengwei Yin^{1, 2} and
 Siqing Gan^{1}Email author
https://doi.org/10.1186/s1366201506999
© Yin and Gan 2015
Received: 9 August 2015
Accepted: 16 November 2015
Published: 1 December 2015
Abstract
To solve the stiff stochastic differential equations, we propose an improved Milstein method, which is constructed by adding an error correction term to the Milstein scheme. The correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the Milstein continuoustime extension. The scheme is proved to be strongly convergent with order one and is as easy to implement as standard explicit schemes but much more efficient for solving stiff stochastic problems. The efficiency and the advantage of the method lie in its very large stability region. For a linear scalar test equation, it is shown that the meansquare stability domain of the method is much bigger than that of the Milstein method. Finally, numerical examples are reported to highlight the accuracy and effectiveness of the method.
Keywords
1 Introduction
Stochastic differential equations (SDEs) play a prominent role in a range of scientific areas like biology, chemistry, epidemiology, mechanics, microelectronics, and finance [1–6]. Since explicit solutions are rarely available for nonlinear SDEs, numerical approximations become increasingly important in many applications. To make the implementation viable, effective numerical methods are clearly the key ingredient and deserve much investigation. In the present work we make efforts in this direction and propose a new efficient scheme, which enjoys cheap computational costs in a strong approximation of stiff SDEs.
Over the last decades, much progress has been made in construction and analysis of various numerical schemes for (1.1) from different numerical points of view; see, e.g., [7–26]. Roughly speaking, there are two major types of numerical methods, called explicit numerical methods and implicit numerical methods in the existing literature. As for the deterministic case, explicit methods [7–9] are easy to implement and are advocated to solve nonstiff problems. For stiff problems, however, the standard explicit methods with poor stability properties suffer a lot from stepsize reduction and turn out to be inefficient in terms of overall computational costs. In order to address this issue, a number of implicit methods including driftimplicit methods [10, 15] and fully implicit methods [12, 16–19, 21, 22] have been introduced, which possess better stability properties than the explicit methods and thus are well adapted for stiff problems. Although implicit methods can usually ease the difficulty arising from stiffness in SDEs, one needs to solve (possibly large) nonlinear algebraic equations at each time step. This might lead to traditional implicit methods still being costly when they are used to approximate large stiff systems, for instance, SDEs produced from spatially discretization of stochastic partial differential equations. In this paper, an improved Milstein (IM) method is developed, which successfully avoids solving nonlinear algebra equations encountered with implicit methods as mentioned above. More importantly, the proposed scheme admits good meansquare stability (MSstability) properties and therefore serves as a good candidate to treat stiff SDEs.
In addition, it is worthwhile to point out that the proposed scheme is close to the Rosenbrock type methods in the literature [24, 27] due to the presence of the inverse Jacobian matrices. Despite the similarity, the scheme we develop here does not coincide with any Rosenbrock type method formulated in [24, 27]. Indeed, the new scheme can be regarded as a modified version of the predictorcorrector method. Based on the classical Milstein method as a predictor, a corrector term involved with the inverse Jacobian matrices is determined following the way in Section 2 of the manuscript. This approach is different from the idea of obtaining Rosenbrock type methods in the literature. Similarly to Rosenbrock methods, the proposed scheme is well suited for stiff problems with the stiffness itself appearing in linear terms from the drift coefficients, and may lose efficiency in the case of stiff nonlinear terms. Finally, we would like to mention that the idea of error correction methods based on Chebyshev collocation was previously employed in [28] to construct methods for stiff deterministic ordinary differential equations.
The remainder of the paper is organized as follows. In the next section, how to construct the IM scheme based on the local truncated error analysis is presented. In Section 3, the strong convergence order in meansquare sense is analyzed. Section 4 is devoted to the MSstability of the IM method. Numerical experiments are reported to confirm the accuracy and effectiveness of the method in Section 5. At the end of this article, conclusions are made briefly.
2 Derivation of the IM method
3 Meansquare convergence analysis
In this section, we will justify the proposed method by proving its strong convergence order of one in meansquare sense. To this end, we make the following standard assumptions [15].
Assumption 3.1
 (1)(globally Lipschitz condition) for all \(x,y\in\mathbb{R}\)$$ \biglf(x)f(y)\bigr^{2}\vee\biglg(x)g(y)\bigr^{2}\leq Lxy^{2}, $$(3.1)
 (2)(linear growth condition) for all \(x\in\mathbb{R}\)$$ \biglf(x)\bigr^{2}\vee\biglg(x)\bigr^{2}\leq K \bigl(1+x^{2}\bigr). $$(3.2)
Here and throughout this work, we use the convention that K represents a generic positive constant independent of h, whose value may be different for different appearances. This assumption guarantees the existence and uniqueness of the exact solution \(X(t)\) of equation (2.1), and, moreover, the solution \(X(t)\) satisfies \(\sup_{0\leq t\leq T} \mathbb{E}X(t)^{2}<\infty\); see, e.g., [30] for more details. In addition, we require the following assumption.
Assumption 3.2
Assume that the functions \(f(x)\) and \(g(x)\) in (2.1) have continuously bounded derivatives up to the required order for the following analysis, and the coefficient functions in ItôTaylor expansions (up to a sufficient order) are globally Lipschitz and satisfy the linear growth conditions.
Subsequently, we present the fundamental strong convergence theorem [29, 31], which was frequently used in the literature to establish the meansquare convergence orders of various numerical schemes.
Theorem 3.3
The notations used in Theorem 3.3 are explained as follows: \(Y_{k}\) generated by the onestep method is an approximation to the exact solution \(X(t_{k})\) of (1.1) with \(t_{k}=kh\), \(X_{t,x}(t+h)\) denotes the exact solution of (1.1) with initial value x at time t and \(\bar{X}_{t,x}(t+h)\) denotes a numerical solution generated by the onestep method with initial value x at time t.
After the above preparations, we start to prove rigorously that the IM method is meansquare convergent with order one under Assumption 3.1 and Assumption 3.2. For simplicity of presentation, we focus on the scalar SDE and the extension to multidimensional case is an easy work and hence omitted here.
Theorem 3.4
Proof
The proof is divided into two steps.
Now an application of Theorem 3.3 with \(p_{1}=2\) and \(p_{2}=\frac{3}{2}\) shows that the scheme is meansquare convergent with order \(p_{2}\frac{1}{2}=1\). □
In the same manner, it is not hard to establish the meansquare convergence of order one for IM method (2.17) applied to general system (1.1).
4 Meansquare stability
For SDEs, two very natural, but distinct stability concepts are MSstability and asymptotical stability. MSstability is applied to measure the stability of moments, while the asymptotical stability is to measure the overall behavior of sample function. In this section, we focus on the MSstability of the IM method applied to a scalar linear test equation.
Definition 4.1
Theorem 4.2
Proof
It turns out that the proposed scheme is not meansquare Astable in the sense that its meansquare stability domain does not contain the stability domain of the exact solution (compare (4.3) and (4.6)). Thus, the stability condition of Theorem 4.2 is not very convenient in practical applications. As immediate consequences of Theorem 4.2, the following corollaries provide convenient stability conditions.
Corollary 4.3
Let \(\lambda, \mu\in\mathbb{R}\) such that \(2 \lambda+ \sqrt{2} \mu^{2}<0\). Then the test problem (4.1) is MSstable and the proposed method is MSstable for any step size \(h >0\).
Based on the above observations, we believe that the new scheme is well suited for stiff meansquare stable problems with moderate (small) stochastic noise intensity or additive noise, where the drift coefficient plays an essential role in the dynamics.
Corollary 4.4
Suppose that \(2\operatorname{Re}\lambda+\mu^{2}<0\) and \(\operatorname{Re}\lambda\leq \operatorname{Im}\lambda\), then the IM method is MSstable for any step size \(h>0\).
Proof
Remark 4.5
5 Numerical tests
Example 1

parameter I: \(\lambda=2\), \(\mu=1\),

parameter II: \(\lambda=20\), \(\mu=5\).
Meansquare errors for ( 5.1 ) with \(\pmb{\lambda=2}\) , \(\pmb{\mu=1}\)
h  \(\boldsymbol {h=2^{5}}\)  \(\boldsymbol {h=2^{6}}\)  \(\boldsymbol {h=2^{7}}\)  \(\boldsymbol {h=2^{8}}\)  \(\boldsymbol {h=2^{9}}\) 

IM  0.8286  0.3975  0.1944  0.0979  0.0485 
Milstein  1.4909  0.7991  0.4136  0.2082  0.1053 
Meansquare errors for ( 5.1 ), \(\pmb{\lambda=20}\) , \(\pmb{\mu=5}\)
h  \(\boldsymbol {2^{1}}\)  \(\boldsymbol {2^{2}}\)  \(\boldsymbol {2^{3}}\)  \(\boldsymbol {2^{4}}\) 

IM  0.71000  0.3733  0.0669  0.0004 
Milstein  1.738e+002  1.709e+004  1.033e+004  3.347e+002 
In order to offer further insight into the above stability results, we restrict ourselves to \(\lambda,\mu\in\mathbb{R}\) in (5.1) and plot MSstability regions of the IM and Milstein methods in Figure 1. As shown there, the MSstability region of the IM method is much larger than that of the Milstein method.
Example 2
Example 3
Meansquare errors of the numerical solutions for ( 5.4 ) with \(\pmb{\lambda=2}\)
h  \(\boldsymbol {h=2^{3}}\)  \(\boldsymbol {h=2^{4}}\)  \(\boldsymbol {h=2^{5}}\)  \(\boldsymbol {h=2^{6}}\) 

IM  0.0073  0.0038  0.0020  0.0012 
Milstein  0.0100  0.0048  0.0025  0.0014 
Relative errors of the numerical solutions for ( 5.4 ) with \(\pmb{\lambda=2}\)
h  \(\boldsymbol {h=2^{3}}\)  \(\boldsymbol {h=2^{4}}\)  \(\boldsymbol {h=2^{5}}\)  \(\boldsymbol {h=2^{6}}\) 

IM  0.0037  0.0019  9.708e−004  5.567e−004 
Milstein  0.0057  0.0027  0.0014  7.187e−004 
Meansquare errors of the numerical solutions for ( 5.4 ) with \(\pmb{\lambda=15}\)
h  \(\boldsymbol {h=2^{6}}\)  \(\boldsymbol {h=2^{7}}\)  \(\boldsymbol {h=2^{8}}\)  \(\boldsymbol {h=2^{9}}\) 

IM  0.0117  0.0017  7.113e−004  7.424e−004 
Milstein  0.0183  0.0126  0.0068  0.0016 
Relative errors of the numerical solutions for ( 5.4 ) with \(\pmb{\lambda=15}\)
h  \(\boldsymbol {h=2^{6}}\)  \(\boldsymbol {h=2^{7}}\)  \(\boldsymbol {h=2^{8}}\)  \(\boldsymbol {h=2^{9}}\) 

IM  0.0056  0.0011  8.272e−004  0.0012 
Milstein  0.1039  0.0148  0.0066  0.0024 
6 Conclusions
This work has proposed the IM method for solving stiff stochastic differential equations. The method is derived by adding a correction term to the classical Milstein method and is easy to implement. Further, the good MSstability and strong convergence order of one are obtained for the scheme. It turns out that the IM method has a larger MSstability region than the classical Milstein method. Numerical results also confirm that the IM method is computationally effective and superior to the Milstein method for solving stiff SDE systems.
In this work, we always assume that the drift and diffusion functions satisfy globally Lipschitz conditions (cf. (3.1)), which excludes many important model equations in applications. Therefore, a future direction is to establish strong convergence rate of the IM scheme for SDEs under a nonglobal Lipschitz condition as studied in [34].
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their valuable and insightful comments which have improved the paper. This work is supported by the National Natural Science Foundation of China (No. 11171352, No. 11571373, No. 11371123), the New Teachers’ Specialized Research Fund for the Doctoral Program from Ministry of Education of China (No. 20120162120096) and Mathematics and Interdisciplinary Sciences Project, Central South University.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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