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On Lévy and Markov additive friendships

Lukas Trottner
Stochastics Seminar
Wednesday, 1 December, 2021 | 13:15–14:00 | Aud. D2 (1531-119)
Contact: Claudia Strauch, Andreas Basse-O'Connor and Rodrigo Labouriau

Vigon's theory of friendship and philanthropy deals with necessary and sufficient criteria for the characteristic exponents of two subordinators (friends) to be the factors in the spatial Wiener–Hopf factorization of a Lévy process. From an analytic perspective, his results yield a rich class of pairs of continuous negative definite functions that are stable under multiplication. We give a probabilistic meaning to this analytic property by investigating uniqueness of the Wiener–Hopf factorization, thus showing that friends can be interpreted as ladder height processes of the factorized Lévy process. Furthermore, we extend Vigon's theory of friendship and philanthropy to Markov additive processes (MAPs) by making use of the matrix Wiener–Hopf factorization of MAP exponents. This allows us to construct MAPs with explicit ladder height processes, adding to the only other known example of such a factorization, which is Kyprianou's deep factorization of the stable process.

Organised by: Stochastics Group