The direct Richardson \(p\)th order (DRp) schemes: a new class of time integration schemes for stochastic differential equations (Q2882782)

The authors introduce a new family of weak approximation methods called direct Richardson \(p\)th-order (DRp) schemes for the class of vector-valued stochastic differential equations NEWLINE\[NEWLINEd X = D(X,t)d t+ \sigma(X,d)dW,NEWLINE\]NEWLINE where \(W\) is a standard vector-valued Wiener process. The new schemes perform Richardson extrapolation on the Euler method in each time step by means of an acceptance-rejection algorithm. The step-by-step implementation of the algorithms is explained in detail. The effective difference to previously published weak approximation methods using Richardson extrapolation is that those are only accurate of the proposed order at the final time value, whereas as the new class of methods produces accurate approximations at every grid point.NEWLINENEWLINEThe main mathematical result of the paper is the proof of the weak convergence of the DRp-schemes of the proposed order under the assumption that the drift vector field and the diffusion matrix are smooth and satisfy standard boundedness and Lipschitz conditions. Further, the authors numerically compare the efficiency of the DRp-schemes to other weak methods on two test cases. It is concluded that the efficiency is overall comparable, i.e., implementations are slightly more expensive but produce a smaller error. Hence, the new class of methods presents an alternative which could possess advantages for specific applications, for example, as the authors mention, if the decomposition of the diffusion matrix is known.