TY - JOUR
T1 - Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon
AU - Bladt, Mogens
AU - Ivanovs, Jevgenijs
N1 - Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/12
Y1 - 2021/12
N2 - There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.
AB - There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.
KW - Functions of matrices
KW - Rational Laplace transform
KW - Scale function
KW - Wiener–Hopf factorization
UR - http://www.scopus.com/inward/record.url?scp=85114134362&partnerID=8YFLogxK
U2 - 10.1016/j.spa.2021.08.002
DO - 10.1016/j.spa.2021.08.002
M3 - Journal article
AN - SCOPUS:85114134362
SN - 0304-4149
VL - 142
SP - 105
EP - 123
JO - Stochastic Processes and Their Applications
JF - Stochastic Processes and Their Applications
ER -