A multitype continuous-state branching process (MCSBP) ${ \rm Z}=({ \rm Z}_{ t})_{ t\geq 0}$, is a Markov process with values in $[0,\infty)^{ d}$ that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of $d$ Laplace exponents of $\mathbb{ R}^d$-valued spectrally positive L\'evy processes, each one having $d-1$ increasing components. We give an expression of the probability for a MCSBP to tend to 0 at infinity in term of its branching mechanism. Then we prove that this extinction holds at a finite time if and only if some condition bearing on the branching mechanism holds. This condition extends Grey's condition that is well known for $d=1$. Our arguments bear on elements of fluctuation theory for spectrally positive additive L\'evy fields recently obtained in \cite{ cma1} and an extension of the Lamperti representation in higher dimension proved in \cite{ cpgub}.
Categories: math.PR