A Posteriori Error Control and Adaptivity for the IMEX BDF2 Method for PIDEs with Application to Options Pricing Models

摘要

Abstract In this paper, we study a posteriori error estimates of the IMEX BDF2 scheme for time discretizations of solving parabolic partial integro-differential equations, which describe the jump-diffusion option pricing model in finance. Because of the initial singularities of the solution which is due to the nonsmoothness of the payoff function, a posteriori error control and adaptivity will be crucial in solving numerically this type of equations. To derive optimal order a posteriori error estimates, quadratic reconstructions for the IMEX BDF2 method are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Based on these a posteriori error estimates, we further develop a time adaptive algorithm. The numerical implementations are performed with both nonuniform partitions and adaptivity in time. The adaptive algorithm reduce the computational cost substantially and provides efficient error control for the solution.