Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate Abstract   Although some implicit numerical procedures have been developed to treat high nonlinearity, the question whether one can use explicit schemes to achieve convergence rate similar to that of Milsteinxe2x80x99s procedure remained open. This brings us to the current work that focuses on numerical solutions of stochastic differential equations using explicit schemes. Our main goals are to obtain order one convergence in the second moment in a finitetime interval. In contrast to the implicit schemes, explicit schemes are advantageous, easily implementable, and computationally less intensive. To overcome the difficulties due to superlinear growth of the coefficients, a truncation device is used in our algorithm. In addition to reaching aforementioned goals in the analysis part, numerical examples are provided to demonstrate our results.

On parareal algorithms for semilinear parabolic Stochastic PDEs Parareal algorithms are studied for semilinear parabolic stochastic partial differential equations. These algorithms proceed as twolevel integrators, with fine and coarse schemes, and have been designed to achieve a parallel in real time implementation. In this work, the fine integrator is given by the exponential Euler scheme. Two choices for the coarse integrator are considered: the linear implicit Euler scheme, and the exponential Euler scheme. The influence on the performance of the parareal algorithm, of the choice of the coarse integrator, of the regularity of the noise, and of the number of parareal iterations, is investigated , with theoretical analysis results and with extensive numerical experiments.