Reflected spectrally negative stable processes and their governing equations
Research output: Contribution to journal › Research article › Contributed › peer-review
Contributors
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.
Details
Original language | English |
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Pages (from-to) | 227-248 |
Number of pages | 22 |
Journal | Transactions of the American Mathematical Society |
Volume | 368 |
Issue number | 1 |
Publication status | Published - 20 Apr 2015 |
Peer-reviewed | Yes |
Keywords
ASJC Scopus subject areas
Keywords
- Cauchy problem, Fractional derivative, Markov process, Reflecting boundary condition, Stable process