Reflected Spectrally Negative Stable Processes and their Governing Equations

TL;DR

This paper derives the forward equation for a reflected spectrally negative stable process with index greater than one, involving fractional derivatives, and applies numerical methods to compute its transition densities, with implications for fractional Cauchy problems.

Contribution

It explicitly computes transition densities and derives the governing fractional differential equation for reflected spectrally negative stable processes.

Findings

Derived the forward equation involving Riemann-Liouville fractional derivatives.

Developed numerical methods for transition density computation.

Connected results to fractional Cauchy problems.

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.