An Euler scheme approximation enables one to discretize virtually any jump diffusion process in a PIDE or Monte Carlo simulation. It is simply a first order Taylor approximation. It is well known that the discretization error increases as soon as we use a step different from a critical value because of round off error effect.An improvement of this method is to use a predictor corrector method. Much has been said about this method for Monte Carlo simulation for diffusion but very little for PIDE and jump diffusions processes. The contribution of this paper is to examine the Predictor Corrector Method for jump diffusion both for PIDE and Monte Carlo simulations.

In the case of the PIDE, the initial scheme results in a Crank-Nickolson for the diffusion and an explicit Euler scheme for the jump part. We show that we obtain a fourth order convergence rate for the jump part. We also show that we can improve the accuracy error using a Runge Kutta method and an extension of the Simpson's rule. We also present two methods for computing the jump term: the traditional FFT and a straight integration. We discuss the stability of the initial numerical scheme (see [2]). In the case of the Monte Carlo, we show that the convergence rate (for strong and weak convergence, see [3] and [4]) improves only for the local volatility term. We provide numerical results for a specific model, namely the AA model. This model is simply a local volatility model with a Poisson jump (see [1]). This enables us to favorably compare our method to existing ones and show that our method is 7 times faster or has a precision increased by 50 times.

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