A Multi-step Scheme based on Cubic Spline for solving Backward Stochastic Differential Equations
For researchers solving backward stochastic differential equations, this work offers an incremental improvement in numerical stability and accuracy over existing multi-step methods.
The authors propose a multi-step scheme using cubic spline interpolation instead of Lagrange polynomials to solve backward stochastic differential equations, achieving better stability and accuracy. Numerical examples show high accuracy and stability.
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, where Lagrange interpolating polynomials are used to approximate the time-integrands with given values of these integrands at chosen multiple time levels. For a better stability and the admission of more time levels we investigate the application of spline instead of Lagrange interpolating polynomials to approximate the time-integrands. The resulting scheme is a semi-discretization in the time direction involving conditional expectations, which can be numerically solved by using the Gaussian quadrature rules and polynomial interpolations on the spatial grids. Several numerical examples including applications in finance are presented to demonstrate the high accuracy and stability of our new multi-step scheme.