Asymptotic high-order schemes for integro-differential problems arising in markets with jumps

In this paper we deal with the numerical approximation of integro-differential equations arising in financial applications in which jump processes act as the underlying stochastic processes. Our aim is to find finite differences schemes which are high-order accurate for large time simulations. Therefore, we study the asymptotic time behavior of such equations and we define as {\it asymptotic high-order schemes} those schemes that are consistent with this behavior. Numerical tests are presented to investigate the efficiency and the accuracy of such approximations.