Closure approximation error in the mean solution of stochastic differential equations by the hierarchy method

1979 ◽  
Vol 21 (2) ◽  
pp. 181-189 ◽  
Author(s):  
G. Adomian ◽  
K. Malakian
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximation Error ◽  
Closure Approximation ◽  
The Mean

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

Strong Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Mean Square ◽  
Nonlinear Stochastic Differential Equations ◽  
Mean Square Convergence ◽  
The Mean

AbstractIn this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

Analysis of the Closure Approximation for a Class of Stochastic Differential Equations

2017 ◽  
Vol 10 (2) ◽  
pp. 299-330
Author(s):  
Yunfeng Cai ◽  
Tiejun Li ◽  
Jiushu Shao ◽  
Zhiming Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Open Systems ◽  
Numerical Study ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
Exact Result ◽  
Moment Closure ◽  
Closure Approximation ◽  
The Moment

AbstractMotivated by the numerical study of spin-boson dynamics in quantum open systems, we present a convergence analysis of the closure approximation for a class of stochastic differential equations. We show that the naive Monte Carlo simulation of the system by direct temporal discretization is not feasible through variance analysis and numerical experiments. We also show that the Wiener chaos expansion exhibits very slow convergence and high computational cost. Though efficient and accurate, the rationale of the moment closure approach remains mysterious. We rigorously prove that the low moments in the moment closure approximation of the considered model are of exponential convergence to the exact result. It is further extended to more general nonlinear problems and applied to the original spin-boson model with similar structure.

scholarly journals Euler-type approximation for systems of stochastic differential equations

1989 ◽  
Vol 2 (4) ◽  
pp. 239-249 ◽  
Author(s):  
J. Golec ◽  
G. Ladde
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Taylor Formula ◽  
Time Varying ◽  
Stochastic Version ◽  
Type Approximation ◽  
Mean Square Convergence ◽  
The Mean ◽  
Time Invariant

By developing a stochastic version of the Taylor formula, the mean-square convergence of the Euler-type approximation for the solution of systems of Itô-type stochastic differential equations is investigated. Sufficient conditions are given to obtain time-varying and time-invariant error estimates.

scholarly journals A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Mean Square ◽  
Variable Delays ◽  
Mean Square Stability ◽  
The Mean ◽  
Random Jump

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

scholarly journals Implicit Numerical Solutions for Solving Stochastic Differential Equations with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Implicit Methods ◽  
Mean Square Stability ◽  
Test Equation ◽  
The Mean ◽  
Taylor Method ◽  
Strong Order ◽  
Linear Scalar

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.

scholarly journals The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Weijun Zhan ◽  
Qian Guo ◽  
Yuhao Cong
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Convergence Result ◽  
Convergence Order ◽  
Mean Square ◽  
Numerical Example ◽  
Milstein Method ◽  
The Mean

<p style='text-indent:20px;'>In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.</p>

Itô-Taylor Schemes for Solving Mean-Field Stochastic Differential Equations

2017 ◽  
Vol 10 (4) ◽  
pp. 798-828 ◽  
Author(s):  
Yabing Sun ◽  
Jie Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Taylor Expansion ◽  
Mean Field ◽  
Strong Convergence Theorem ◽  
The Mean ◽  
New Formula ◽  
Strong Order ◽  
Taylor Scheme ◽  
Theoretical Results

AbstractThis paper is devoted to numerical methods for mean-field stochastic differential equations (MSDEs). We first develop the mean-field Itô formula and mean-field Itô-Taylor expansion. Then based on the new formula and expansion, we propose the Itô-Taylor schemes of strong order γ and weak order η for MSDEs, and theoretically obtain the convergence rate γ of the strong Itô-Taylor scheme, which can be seen as an extension of the well-known fundamental strong convergence theorem to the mean-field SDE setting. Finally some numerical examples are given to verify our theoretical results.

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