Numerical analysis of the balanced implicit methods for stochastic pantograph equations with jumps

Hu, Luojia; Gan, Siqing

doi:10.1016/j.amc.2013.08.021

Cited by 9 publications

(6 citation statements)

References 8 publications

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“…where ( ) is F -adapted Wiener process and ( ) is a scalar poisson process with intensity and is independent of ( ). Hu and Gan [22,25] proposed the balanced method for SDEJs (1) and stochastic pantograph equations with jumps, respectively, and proved that the numerical solution converges to the analytical solution with rate 1/2. The asymptotic stability of the balanced method for SDEJs (1) was obtained in [26].…”

Section: Introductionmentioning

confidence: 99%

“…The other is the fully implicit methods in which both the drift components and the diffusion components are computed implicitly. Since implicit stochastic terms in the implicit methods lead to infinite absolute moments of the numerical solution, extensive research has been done to address this issue [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. For example, Milstein et al [11] proposed the balanced implicit method for the numerical solutions of SDEs.…”

Section: Introductionmentioning

confidence: 99%

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Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

Mei

2014

Abstract and Applied Analysis

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To realize the applications of stochastic differential equations with jumps, much attention has recently been paid to the construction of efficient numerical solutions of the equations. Considering the fact that the use of the explicit methods often results in instability and inaccurate approximations in solving stochastic differential equations, we propose two implicit methods, theθ-Taylor method and the balancedθ-Taylor method, for numerically solving the stochastic differential equation with jumps and prove that the numerical solutions are convergent with strong order 1.0. For a linear scalar test equation, the mean-square stability regions of the methods are derived. Finally, numerical examples are given to evaluate the performance of the methods.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

Mei

2014

Abstract and Applied Analysis

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“…Therefore, there are various numerical methods developed for their solution -cf. [2,3,11,13,25,[28][29][30]. In the last decades, the numerical methods preserving the geometric invariants along the flows, such as symplectic and Lie group structures, and quantities showing that the exact solution evolves on a manifold of the dimension smaller than that of n , have been widely studied [5,8,9,27,32].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

Wang¹,

Ma²,

Ding³

2022

EAJAM

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The deterministic discrete gradient method for stochastic differential equations is extended to equations with multiple conserved quantities. The equations with multiple conserved quantities in the Stratonovich sense are written in the skew-gradient form, which is used in the construction of the stochastic discrete gradient method. It is shown that the stochastic discrete gradient method has the mean-square convergence order one and preserves all conserved quantities. Besides, for a given skew-gradient form, the stochastic discrete gradient method is equivalent to the stochastic projection method. Numerical examples confirm the theoretical results and show the effectiveness of the method.

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“…Therefore, scholars have paid their attention to construct efficient numerical methods to simulate the solution of SDEs. In the last decades, many aspects of theory of numerical methods for SDEs are investigated (see, for example, in [2][3][4][5][6][7][8][9][10] and references therein). Generally, numerical methods for SDEs can be classified to three categories: (1) Explicit methods which are easy to implement but only suitable for non-stiff problems [11,12];…”

Section: Introductionmentioning

confidence: 99%

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

Wang

Ding

2020

Mathematics

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Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.

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Numerical analysis of the balanced implicit methods for stochastic pantograph equations with jumps

Cited by 9 publications

References 8 publications

Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

Numerical Simulations of Stochastic Differential Equations with Multiple Conserved Quantities by Conservative Methods

Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods

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