Asymptotic mean-square stability of linear multi-step methods for SODEs

PAMM ◽  
2006 ◽  
Vol 6 (1) ◽  
pp. 659-660
Author(s):  
Evelyn Buckwar ◽  
Renate Winkler
Keyword(s):  
Mean Square ◽  
Mean Square Stability ◽  
Multi Step Methods

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

EXPONENTIAL MEAN SQUARE STABILITY OF STOCHASTICALLY FORCED INVARIANT MANIFOLDS FOR NONLINEAR SDEs

2007 ◽  
Vol 07 (03) ◽  
pp. 389-401 ◽  
Author(s):  
L. B. RYASHKO
Keyword(s):  
Linear Extension ◽  
Invariant Manifolds ◽  
Function Technique ◽  
Nonlinear Stochastic System ◽  
General Notion ◽  
Mean Square ◽  
Extension System ◽  
Mean Square Stability ◽  
The Stability ◽  
Exponential Mean Square Stability

An exponential mean square stability for the invariant manifold [Formula: see text] of a nonlinear stochastic system is considered. The stability analysis is based on the [Formula: see text]-quadratic Lyapunov function technique. The local dynamics of the nonlinear system near manifold is described by the stochastic linear extension system. We propose a general notion of the projective stability (P-stability) and prove the following theorem. The smooth compact manifold [Formula: see text] is exponentially mean square stable if and only if the corresponding stochastic linear extension system is P-stable.

Stochastic Stability of Mechanical Systems Under Renewal Jump Process Parametric Excitation

2005 ◽  
Vol 72 (2) ◽  
pp. 213-221 ◽  
Author(s):  
R. Iwankiewicz ◽  
S. R. K. Nielsen ◽  
J. W. Larsen
Keyword(s):  
State Space ◽  
Parametric Excitation ◽  
Stochastic Equation ◽  
Jump Process ◽  
The State ◽  
Arrival Process ◽  
Mean Square ◽  
Direct Simulation ◽  
Excitation Process ◽  
Mean Square Stability

A dynamic system under parametric excitation in the form of a non-Erlang renewal jump process is considered. The excitation is a random train of nonoverlapping rectangular pulses with equal, deterministic heights. The time intervals between two consecutive jumps up (or down), are the sum of two independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However, due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean-square stability condition is not overly conservative.

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