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A method is proposed for the numerical solution of Ito stochastic differential equations by means of a second-order Runge-Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge-Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.It was observed by Wright [8] that different iterative schemes for the numerical solution of stochastic differential equationswhere £, is a Wiener process, converge to different solutions for the same noise sample and initial condition. This is in contrast to their deterministic counterparts for ordinary differential equations, which converge to the same solution.Strictly speaking stochastic differentia] equations (1) are really integral equationsx t = x 0 +\ a(s,x s )ds+\ b(s,x s )d£ s ,[2]Stochastic differential equations 9

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Finally, various background definitions and results needed within the text are given in the appendix. Readers who are interested in the dynamical behavior of nonautonomous partial differential equations and evolution equations are advised to refer to the monographs of Carvalho, Langa & Robinson [35] and Chepyzhov & Vishik [43] in conjunction with this book.

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On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler's approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.