Abstract
A spectrally positive additive Levy field is a multidimensional field obtained as the sum $\mathbf {X}_{\mathrm {t}}={\mathrm {X}}^{(1)}_{t_{1}}+{\mathrm {X}}^{(2)}_{t_{2}}+\dots +{\mathrm {X}}^{(d)}_{t_{d}}$, ${\mathrm {t}}=(t_{1},\dots ,t_{d})\in \mathbb {R}_{+}^{d}$, where ${\mathrm {X}}^{(j)}={}^{t} (X^{1,j},\dots ,X^{d,j})$, $j=1,\dots ,d$, are $d$ independent $\mathbb {R}^{d}$-valued Levy processes issued from $\mathbf {0}={}^{t}(0,0,\dots ,0)$, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf {X}_{\mathrm {t}}=\mathbb {X}_{\mathrm {t}}\cdot {\mathbf {1}}$, where ${\mathbf {1}}={}^{t}(1,1,\dots ,1)$ and $\mathbb {X}_{\mathrm {t}}=(X^{i,j}_{t_{j}})_{1\leq i,j\leq d}$. 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 $\mathbf {T}_{\mathrm {r}}$ of such fields at levels $-{\mathrm {r}}$, where ${\mathrm {r}}\in \mathbb {R}_{+}^{d}$. We prove that the field $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf {X}$. In particular, the Laplace exponent of $(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf {X}$. 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 $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ in terms of the distribution of $\{\mathbb {X}_{\mathrm {t}},{\mathrm {t}}\in \mathbb {R}_{+}^{d}\}$ by the means of a Kemperman-type formula, well-known for spectrally positive Levy processes.
Highlights
A spectrally positive, additive Lévy field is defined by Xt := dXtij,j, i = 1, . . . , d j=1= X(t11) + · · · + X(tdd), t = (t1, . . . , td) ∈ Rd+, where X(j) = t(X1,j, . . . , Xd,j), j = 1, . . . , d, are d independent Rd-valued Lévy processes such that Xi,j are non decreasing for i = j and Xj,j is spectrally positive
We study the law of the first passage times Tr of such fields at levels −r, where r ∈ Rd+
We prove that the field {(Tr, XTr ), r ∈ Rd+} has stationary and independent increments and we describe its law in terms of this of the spaLf X
Summary
D, j=1 are d independent Zd-valued random walks such that Si,j are non decreasing for i = j and Sj,j is downward skip free These random fields are the discrete time counterparts of spaLf’s which suggests the possibility of coding continuous multitype branching trees in an analogous way. 1≤i,j≤d up to its first passage time Tr at level −r This fact which is plain in the case d = 1 will be proved in the general case in the upcoming paper [6], where extinction of continuous time multitype branching processes is characterized through path properties of spaLf’s. This equation, see (4.18) in Theorem 4.1 below, can be compared to the classical Wiener-Hopf factorization involving the ladder processes of spectrally positive Lévy processes. In order to prove it, we will use a similar identity recently obtained in [7] and [5] in the discrete time and space settings together with a discrete approximation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
