The applications of partial integro-differential equations related to adaptive wavelet collocation methods for viscosity solutions to jump-diffusion models

This paper presents adaptive wavelet collocation methods for the numerical solutions to partial integro-differential equations (PIDEs) arising from option pricing in a market driven by jump-diffusion process. The first contribution of this paper lies in the formulation of the wavelet collocation schemes: the integral and differential operators are formulated in the collocation setting exactly and efficiently in both adaptive and non-adaptive wavelet settings. The wavelet compression technique is employed to replace the full matrix corresponding to the nonlocal integral term by a sparse matrix. An adaptive algorithm is developed, which automatically obtains the solution on a near-optimal grid. The second contribution of this paper is the theoretical analysis of the wavelet collocation schemes: due to the possible degeneracy of the parabolic operators, classical solutions of the jump-diffusion models may not exist. In this paper we first prove the convergence and stability of the proposed numerical schemes under the framework of viscosity solution theory, and then the numerical experiments demonstrate the accuracy and computational efficiency of the methods we developed.