Fluctuation theory for spectrally positive additive Lévy fields

Abstract

A spectrally positive additive Lévy 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 Lévy 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 Lévy processes.