This paper presents a strong predictor-corrector method for the numerical solution of stochastic delay differential equations （SDDEs） of ItS-type. The method is proved to be mean-square convergent of order min{1/2,p} under the Lipschitz condition and the linear growth condition, where p is the exponent of HSlder condition of the initial function. Stability criteria for this type of method are derived. It is shown that for certain choices of the flexible parameter p the derived method can have a better stability property than more commonly used numerical methods. That is, for some p, the asymptotic MS-stability bound of the method will be much larger than that of the Euler-Maruyama method. Numerical results are reported confirming convergence properties and comparing stability properties of methods with different parameters p. Finally, the vectorised simulation is discussed and it is shown that this implementation is much more efficient.
This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of （k, l）- algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid. The finite- dimensional and infinite-dimensional dissipativity results of （k, /）-algebraically stable Runge-Kutta methods are obtained.
Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies.A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian a-stable Levy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by solving a nonlocal Fokker-Planck equation numerically. This allows numerically investigating phenomenological bifurcation, or P-bifurcation, for stochastic differential equations with non-Gaussian Levy noises.
