Stability equivalence between the stochastic dierential delay equations driven by G-Brownian motion and the Euler-Maruyama method

Consider a stochastic differential delay equation driven by G-Brownian motion (G-SDDE) dx(t) = f(x(t), x(t − τ))dt + g(x(t), x(t − τ))dB(t) + h(x(t), x(t − τ))dhBi(t). Under the global Lipschitz condition for the G-SDDE, we show that the G-SDDE is exponentially stable in mean square if and only if for sufficiently small step size, the Euler-Maruyama (EM) method is exponentially stable in mean square. Thus, we can carry out careful numerical simulations to investigate the exponential stability of the underlying G-SDDE in practice, in the absence of an appropriate Lyapunov function. A numerical example is provided to illustrate our results

On stability for numerical approximations of stochastic ordinary differential equations

Stochastic ordinary differential equations (SODE) represent physical phenomena driven by stochastic processes. Like for deterministic differential equations, various numerical schemes are proposed for SODE (see references). We will consider several concepts of stability and connection between them

Stability analysis of fractional-order systems with randomly time-varying parameters

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results

Effects of distributed delays on the stability of structures under seismic excitation and multiplicative noise

The effects of seismic excitation and multiplicative noise (arising from environmental fluctuations) on the stability of a single degree of freedom system with distributed delays are investigated. The system is modelled in the form of a stochastic integro-differential equation interpreted in Stratonovich sense. Both deterministic stability and stochastic moment stability are examined for the system in the absence of seismic excitation. The model is also extended to incorporate effects of symmetric nonlinearity. The simulation of stochastic linear and nonlinear systems are carried out by resorting to numerical techniques for the solution of stochastic differential equations

Stochastic stability of structures under active control with distributed time delays

The pathwise behaviour of a single degree of freedom (SDOF) system with symmetric nonlinearity and distributed delays is investigated under the presence of seismic excitation and multiplicative noise. Besides distributed time delays and finite build-up time of control force are taken into consideration. The system is modelled as stochastic integro-differential equation with exponential type kernels. Interpreting stochastic equations in Stratonovich sense, stochastic stability is analyzed in terms of Lyapunov exponents. Estimates of frequencies with which sample paths of displacement of SDOF system cross certain critical values are also obtained. Studies of stochastic linear and nonlinear systems are carried out by resorting to numerical techniques for the solution of (ordinary) stochastic differential equations