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Reflected spectrally negative stable processes and their governing equations


Authors: Boris Baeumer, Mihály Kovács, Mark M. Meerschaert, René L. Schilling and Peter Straka
Journal: Trans. Amer. Math. Soc. 368 (2016), 227-248
MSC (2010): Primary 60G52, 60J50; Secondary 26A33, 60J22
DOI: https://doi.org/10.1090/tran/6360
Published electronically: April 20, 2015
MathSciNet review: 3413862
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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.


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Additional Information

Boris Baeumer
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
Email: bbaeumer@maths.otago.ac.nz

Mihály Kovács
Affiliation: Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand
Email: mkovacs@maths.otago.ac.nz

Mark M. Meerschaert
Affiliation: Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48824
Email: mcubed@stt.msu.edu

René L. Schilling
Affiliation: Institut für Mathematische Stochastik, Technische Universität Dresden, 01069 Dresden, Germany
Email: rene.schilling@tu-dresden.de

Peter Straka
Affiliation: School of Mathematics and Statistics, The University of New South Wales, Kensington NSW 2052, Australia
Email: p.straka@unsw.edu.au

DOI: https://doi.org/10.1090/tran/6360
Keywords: Stable process, reflecting boundary condition, Markov process, fractional derivative, Cauchy problem
Received by editor(s): February 15, 2013
Received by editor(s) in revised form: August 6, 2013, and October 29, 2013
Published electronically: April 20, 2015
Additional Notes: The third author was partially supported by NSF grants DMS-1025486 and DMS-0803360, and NIH grant R01-EB012079.
Article copyright: © Copyright 2015 American Mathematical Society