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In this work we study a multi-step scheme on time-space grids proposed by W.Zhao et al. [28] for solving backward stochastic differential equations, whereLagrange interpolating polynomials are used to approximate the time-integrandswith given values of these integrands at chosen multiple time levels. For abetter stability and the admission of more time levels we investigate theapplication of spline instead of Lagrange interpolating polynomials toapproximate the time-integrands. The resulting scheme is a semi-discretizationin the time direction involving conditional expectations, which can benumerically solved by using the Gaussian quadrature rules and polynomialinterpolations on the spatial grids. Several numerical examples includingapplications in finance are presented to demonstrate the high accuracy andstability of our new multi-step scheme.

A Multi-step Scheme based on Cubic Spline for solving Backward Stochastic Differential Equations