Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises

2015 ◽  
Vol 258 ◽  
pp. 282-295 ◽  
Author(s):  
Xiao Wang ◽  
Jinqiao Duan ◽  
Xiaofan Li ◽  
Yuanchao Luan
Keyword(s):  
Dynamical Systems ◽  
Numerical Methods ◽  
Exit Time ◽  
Two Dimensional ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Mean Exit Time ◽  
The Mean ◽  
Non Gaussian

Parameter estimation via homogenization for stochastic dynamical systems with oscillating coefficients

2018 ◽  
Vol 18 (03) ◽  
pp. 1850025
Author(s):  
Xinyong Zhang ◽  
Hui Wang ◽  
Yanjie Zhang ◽  
Haokun Lin
Keyword(s):  
Parameter Estimation ◽  
Dynamical Systems ◽  
Exit Time ◽  
Least Square Method ◽  
Least Square ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Mean Exit Time ◽  
Oscillating Coefficients ◽  
The Mean

This paper is devoted to studying parameter estimation for a class of stochastic dynamical systems with oscillating coefficients. We show that the homogenized systems faithfully capture the dynamical quantities such as mean exit time and escape probability. Exacting data from observations on the mean exit time (or escape probability) of the original systems, we try to fit the mean exit time (or escape probability) of the homogenized systems by least square method. In this way, we can accurately estimate the unknown parameter in the drift under appropriate assumptions. Furthermore, we conduct some numerical experiments to illustrate our method.

scholarly journals MEAN EXIT TIME AND ESCAPE PROBABILITY FOR A TUMOR GROWTH SYSTEM UNDER NON-GAUSSIAN NOISE

2012 ◽  
Vol 22 (04) ◽  
pp. 1250090 ◽  
Author(s):  
JIAN REN ◽  
CHUJIN LI ◽  
TING GAO ◽  
XINGYE KAN ◽  
JINQIAO DUAN
Keyword(s):  
Tumor Growth ◽  
Exit Time ◽  
Laplacian Operator ◽  
Escape Probability ◽  
Tumor Growth Model ◽  
Bifurcation Phenomena ◽  
Mean Exit Time ◽  
Growth System ◽  
The Mean ◽  
Non Gaussian

Effects of non-Gaussian α-stable Lévy noise on the Gompertz tumor growth model are quantified by considering the mean exit time and escape probability of the cancer cell density from inside a safe or benign domain. The mean exit time and escape probability problems are formulated in a differential-integral equation with a fractional Laplacian operator. Numerical simulations are conducted to evaluate how the mean exit time and escape probability vary or bifurcates when α changes. Some bifurcation phenomena are observed and their impacts are discussed.

scholarly journals AN INTERMEDIATE REGIME FOR EXIT PHENOMENA DRIVEN BY NON-GAUSSIAN LÉVY NOISES

2008 ◽  
Vol 08 (03) ◽  
pp. 583-591 ◽  
Author(s):  
ZHIHUI YANG ◽  
JINQIAO DUAN
Keyword(s):  
Dynamical System ◽  
Noise Intensity ◽  
Exit Time ◽  
First Exit Time ◽  
Small Noise ◽  
Intermediate Regime ◽  
Mean Exit Time ◽  
Small Intensity ◽  
The Mean ◽  
Non Gaussian

A dynamical system driven by non-Gaussian Lévy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian Lévy noises, it is shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable Lévy noise case), in terms of the reciprocal of the small noise intensity.

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

2014 ◽  
Vol 36 (3) ◽  
pp. A887-A906 ◽  
Author(s):  
T. Gao ◽  
J. Duan ◽  
X. Li ◽  
R. Song
Keyword(s):  
Dynamical Systems ◽  
Exit Time ◽  
Escape Probability ◽  
Mean Exit Time

Asymptotic methods for stochastic dynamical systems with small non-Gaussian Lévy noise

2014 ◽  
Vol 15 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Huijie Qiao ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Asymptotic Methods ◽  
Lévy Noise ◽  
Stochastic Dynamical Systems ◽  
Escape Probability ◽  
Nonlocal Interactions ◽  
Escape Probabilities ◽  
Non Gaussian

The goal of the paper is to analytically examine escape probabilities for dynamical systems driven by symmetric α-stable Lévy motions. Since escape probabilities are solutions of a type of integro-differential equations (i.e. differential equations with nonlocal interactions), asymptotic methods are offered to solve these equations to obtain escape probabilities when noises are sufficiently small. Three examples are presented to illustrate the asymptotic methods, and asymptotic escape probability is compared with numerical simulations.

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