Reflected Spectrally Negative Stable Processes and their Governing Equations
Abstract
This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.
Keywords
Cite
@article{arxiv.1301.5605,
title = {Reflected Spectrally Negative Stable Processes and their Governing Equations},
author = {Boris Baeumer and Mihály Kovács and Mark M. Meerschaert and René L. Schilling and Peter Straka},
journal= {arXiv preprint arXiv:1301.5605},
year = {2016}
}