Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients

Gan, Siqing; He, Youzi; Wang, Xiaojie

doi:10.1016/j.apnum.2019.11.014

Cited by 24 publications

(4 citation statements)

References 31 publications

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“…When the delay component vanishes, the underlying SDDEs reduce to the classical stochastic differential equations (SDEs), numerical methods for which have been extensively investigated for the past decades under the global Lipschitz condition (see, e.g., [25], [21], [39]). In the setting of SDEs whose coefficients can be allowed to grow super-linearly, several explicit schemes have been introduced, including tamed EM and Runge-Kutta schemes [18,24,40], balanced EM schemes [46,49] and truncated EM schemes [30,32,33]. Recently, the attention of some researches was attracted to the strong convergence of explicit numerical methods for super-linear SDEs with delay, i.e., SDDEs.…”

Section: Introductionmentioning

confidence: 99%

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

Deng¹,

Fei²,

Mao³

2024

JCM

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This work is concerned with the convergence and stability of the truncated Euler-Maruyama (EM) method for super-linear stochastic differential delay equations (SDDEs) with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types of the truncated EM method for such jump-diffusion SDDEs with time-variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The first type is proved to have a strong convergence order which is arbitrarily close to 1/2 in mean-square sense, under the Khasminskii-type, global monotonicity with U function and polynomial growth conditions. The second type is convergent in qth (q < 2) moment under the local Lipschitz plus generalized Khasminskii-type conditions. In addition, we show that the partially truncated EM method preserves the mean-square and H∞ stabilities of the true solutions. Lastly, we carry out some numerical experiments to support the theoretical results.

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Section: Introductionmentioning

confidence: 99%

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

Deng¹,

Fei²,

Mao³

2024

JCM

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“…Moreover, Wang et al [17] proposed the tamed Milstein method for SDEs with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. Also, Gan et al [18] introduced a family of explicit tamed stochastic Runge-Kutta methods for commutative SDEs with superlinearly growing coefficients. However, as noted in [10], the tamed methods can lead to inaccurate results even for moderately small step sizes.…”

Section: Introductionmentioning

confidence: 99%

A split-step Euler-Maruyama based method with partial truncation coefficients for nonlinear SDEs

Haghighi

2024

Preprint

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The main objective of this paper is to develop and analyze a numerical method for the strong approximation of solutions to highly nonlinear stiff It\^{o} stochastic differential equations (SDEs). In this method, we first divide the coefficients of the SDEs into linear and nonlinear parts. Then, in two fully explicit recursive phases, we apply the appropriate truncation transformation only to the nonlinear parts of the coefficients of the SDEs at each step. Theoretical aspects are introduced to establish the $L^q$-convergence theory of the new method in the finite time interval $[0,T]$. The stability and boundedness properties of the solutions generated by the new method are also investigated. An important contribution is that the region of MS-stability of the new method includes the region of MS-stability of the partially truncated method proposed by Yang et al. (2022). Moreover, we show that the proposed method preserves the exponential MS-stability and the asymptotic boundedness of the SDEs. Finally, numerical experiments are performed to compare the efficiency of this method with an existing explicit method developed for nonlinear SDEs. MSC Classification: 65C30 , 65L07 , 65L04

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“…[11][12][13]17]. In particular, the balanced and tamed explicit methods have received much attention from scholars in recent years, for example, the balanced EM method [24], the balanced Milstein method [31], the split-step balanced θ-method [14], the semi-tamed Milstein method [15], the tamed Milstein method [27], and the tamed Runge-Kutta methods [4].…”

Section: Introductionmentioning

confidence: 99%

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

Yu,

Tang

2024

JCM

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In this paper, we introduce a new class of explicit numerical methods called the tamed stochastic Runge-Kutta-Chebyshev (t-SRKC) methods, which apply the idea of taming to the stochastic Runge-Kutta-Chebyshev (SRKC) methods. The key advantage of our explicit methods is that they can be suitable for stochastic differential equations with non-globally Lipschitz coefficients and stiffness. Under certain non-globally Lipschitz conditions, we study the strong convergence of our methods and prove that the order of strong convergence is 1/2. To show the advantages of our methods, we compare them with some existing explicit methods (including the Euler-Maruyama method, balanced Euler-Maruyama method and two types of SRKC methods) through several numerical examples. The numerical results show that our t-SRKC methods are efficient, especially for stiff stochastic differential equations.

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Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients

Cited by 24 publications

References 31 publications

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients

A split-step Euler-Maruyama based method with partial truncation coefficients for nonlinear SDEs

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

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