Implicit–explicit numerical schemes for jump–diffusion processes

Abstract

We study the numerical approximation of solutions for parabolic integro-differential equations (PIDE). Similar models arise in option pricing, to generalize the Black–Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Due to the non-local nature of the integral term, unconditionally stable implicit difference schemes are not practically feasible. Here we propose using implicit-explicit (IMEX) Runge-Kutta methods for the time integration to solve the integral term explicitly, giving higher-order accuracy schemes under weak stability time-step restrictions. Numerical tests are presented to show the computational efficiency of the approximation. Mathematics Subject Classification (1991): Primary: 65M12; Secondary: 35K55, 49L25