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Fluctuation theory for spectrally positive additive Lévy fields

Abstract : A spectrally positive additive Lévy field is a multidimensional field obtained as the sum X t = X (1) t1 + X (2) t2 + · · · + X (d) t d , t = (t 1 ,. .. , t d) ∈ R d + , where X (j) = t (X 1,j ,. .. , X d,j), j = 1,. .. , d, are d independent R d-valued Lévy processes issued from 0, such that X i,j is non decreasing for i = j and X j,j is spectrally positive. It can also be expressed as X t = X t 1, where 1 = t (1, 1,. .. , 1) and X t = (X i,j tj) 1≤i,j≤d. The main interest of spaLf's lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times T r of such fields at levels −r, where r ∈ R d +. We prove that the field {(T r , X Tr), r ∈ R d + } has stationary and independent increments and we describe its law in terms of this of the spaLf X. In particular, the Laplace exponent of (T r , X Tr) solves a functional equation leaded by the Laplace exponent of X. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of {(T r , X Tr), r ∈ R d + } in terms of the distribution of {X t , t ∈ R d + } by the means of a Kemperman-type formula, well-known for spectrally positive Lévy processes.
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Submitted on : Friday, January 3, 2020 - 7:57:33 PM
Last modification on : Wednesday, November 3, 2021 - 9:15:01 AM
Long-term archiving on: : Monday, April 6, 2020 - 8:38:32 PM

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Loïc Chaumont, Marine Marolleau. Fluctuation theory for spectrally positive additive Lévy fields. 2020. ⟨hal-02427665⟩

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