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.

Multilevel Monte Carlo by using the Halton sequence

2020 ◽  
Vol 26 (3) ◽  
pp. 193-203
Author(s):  
Shady Ahmed Nagy ◽  
Mohamed A. El-Beltagy ◽  
Mohamed Wafa
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Computational Cost ◽  
Random Numbers ◽  
Multilevel Monte Carlo ◽  
Halton Sequence ◽  
Random Generator ◽  
Mc Simulation ◽  
Multilevel Monte Carlo Method

AbstractMonte Carlo (MC) simulation depends on pseudo-random numbers. The generation of these numbers is examined in connection with the Brownian motion. We present the low discrepancy sequence known as Halton sequence that generates different stochastic samples in an equally distributed form. This will increase the convergence and accuracy using the generated different samples in the Multilevel Monte Carlo method (MLMC). We compare algorithms by using a pseudo-random generator and a random generator depending on a Halton sequence. The computational cost for different stochastic differential equations increases in a standard MC technique. It will be highly reduced using a Halton sequence, especially in multiplicative stochastic differential equations.

Approximate analytical solutions for a class of nonlinear stochastic differential equations

2018 ◽  
Vol 30 (5) ◽  
pp. 928-944 ◽  
Author(s):  
A. T. MEIMARIS ◽  
I. A. KOUGIOUMTZOGLOU ◽  
A. A. PANTELOUS
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Approximate Analytical Solution ◽  
Transition Probability ◽  
Computational Cost ◽  
Closed Form Expression ◽  
Response Process ◽  
Approximate Analytical Solutions ◽  
Constant Diffusion

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

scholarly journals Numerical Study of Random Periodic Lipschitz Shadowing of Stochastic Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Qingyi Zhan ◽  
Xiangdong Xie
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Numerical Study ◽  
Numerical Approach ◽  
Logistic Equation ◽  
Numerical Implementation ◽  
Mean Square ◽  
Numerical Computations

This paper is devoted to a new numerical approach for the possibility of(ω,Lδ)-periodic Lipschitz shadowing of a class of stochastic differential equations. The existence of(ω,Lδ)-periodic Lipschitz shadowing orbits and expression of shadowing distance are established. The numerical implementation approaches to the shadowing distance by the random Romberg algorithm are presented, and the convergence of this method is also proved to be mean-square. This ensures the feasibility of the numerical method. The practical use of these theorems and the associated algorithms is demonstrated in the numerical computations of the(ω,Lδ)-periodic Lipschitz shadowing orbits of the stochastic logistic equation.

Linear Systems Excited by Polynomials of Filtered Poission Pulses

1997 ◽  
Vol 64 (3) ◽  
pp. 712-717 ◽  
Author(s):  
M. Di Paola
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Linear Systems ◽  
Moment Equations ◽  
Poisson White Noise ◽  
Finite Set ◽  
The Moment ◽  
Memoryless Transformation

The stochastic differential equations for quasi-linear systems excited by parametric non-normal Poisson white noise are derived. Then it is shown that the class of memoryless transformation of filtered non-normal delta correlated process can be reduced, by means of some transformation, to quasi-linear systems. The latter, being excited by parametric excitations, are frst converted into ltoˆ stochastic differential equations, by adding the hierarchy of corrective terms which account for the nonnormality of the input, then by applying the Itoˆ differential rule, the moment equations have been derived. It is shown that the moment equations constitute a linear finite set of differential equation that can be exactly solved.

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