Space-Time Wavelet Methods for degenerate parabolic PDEs with applications to Fractional Brownian motion models

Abstract

We analyze parabolic PDEs with certain type of weakly singular or degenerate time-dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense. For the numerical solution a weak space-time formulation is considered, as a possible singularity or degeneracy of the diffusion coefficients impedes the application of classical parabolic theory. We analyze parabolic PDEs with certain type of weakly singular or degenerate time-dependent coefficients and prove existence and uniqueness of weak solutions in an appropriate sense. For the numerical solution a weak space-time formulation is considered, as a possible singularity or degeneracy of the diffusion coefficients impedes the application of classical parabolic theory. The use of appropriate wavelet bases in the space-time domain leads to Riesz bases for the ansatz and test spaces. Applications to fractional Brownian motion models in option pricing are presented. As pricing problems are typically posed on unbounded spatial domains, a localization for the PDE with different boundary conditions and the arising truncation estimates are presented.