We extend the original Mullins theory of surface grooving due to a single interface to multiple interacting grooves formed due to closely spaced flat interfaces. First, we show that Mullins’ analysis for one groove can be simplified by using Fourier cosine transforms instead of Laplace transforms. Second, we solve the corresponding problem for an infinite periodic row of grooves. For both of th...

In this work, we propose to price Parisian options using Laplace transforms. Not only, do we compute the Laplace transforms of all the different Parisian options, but we also explain how to invert them numerically. We discuss the accuracy of the numerical inversion and present the evolution of the Greeks through a few graphs.

Report II is concerned with the extended results of distance function wavelets (DFW). The fractional DFW transforms are first addressed relating to the fractal geometry and the fractional derivative, and then, the discrete Helmholtz-Fourier transform is briefly presented. The Green second identity may be an alternative devise in developing the theoretical framework of the DFW transform and seri...