Variable step size implementation of the Balanced Milstein method for stochastic differential equations

The balanced-Milstein (BM) method of strong order 1 is introduced based on the idea of the balanced-implicit (BI) method for solving stiff stochastic differential equations. We investigate the implementation of a variable step size for the BI method of strong order 1/2; and also for an embedded pair (BM, BI) method. Numerical experiment shows that variable step size implementation of the BI method does not converges to the correct solution, while the embedded (BM, BI) scheme show convergence to the Ito solution. We also consider an alternative approach by applying Richardsonpsilas extrapolation on the BM method and numerical results show better performance.