Convergence of schemes for stochastic differential equations

In many applications of stochastic calculus, like stochastic dynamical systems, stochastic differential equations are involved, the coefficients of which are not globally, but only locally Lipschitz. For instance, in order to study technics using one trajectory of a process defined by differential equations of oscillators associated to structures submitted to a white noise excitation, such as the random decrement, one need to simulate a trajectory for such a process. Different schemes are proposed to numerically solve such stochastic differential equations: Euler, Milshtein or Newmark schemes for example. In this paper, the almost sure convergence of some of the most important schemes is studied under locally Lipschitz assumptions and a speed of convergence is established.