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We introduce a predictor-corrector discretisation scheme for the numericalintegration of a class of stochastic differential equations and prove that itconverges with weak order 1.0. The key feature of the new scheme is that itbuilds up sequentially (and recursively) in the dimension of the state space ofthe solution, hence making it suitable for approximations of high-dimensionalstate space models. We show, using the stochastic Lorenz 96 system as a testmodel, that the proposed method can operate with larger time steps than thestandard Euler-Maruyama scheme and, therefore, generate valid approximationswith a smaller computational cost. We also introduce the theoretical analysisof the error incurred by the new predictor-corrector scheme when used as abuilding block for discrete-time Bayesian filters for continuous-time systems.Finally, we assess the performance of several ensemble Kalman filters thatincorporate the proposed sequential predictor-corrector Euler scheme and thestandard Euler-Maruyama method. The numerical experiments show that the filtersemploying the new sequential scheme can operate with larger time steps, smallerMonte Carlo ensembles and noisier systems.

Sequential discretisation schemes for a class of stochastic differential equations and their application to Bayesian filtering