Extinction times of multitype continuous-state branching processes

A multitype continuous-state branching process (MCSBP) Z = (Zt)t >= 0, is a Markov process with values in [0, infinity)d that satisfies the branching property. Its distribution is characterised by its branching mechanism, that is the data of d Laplace exponents of Rd-valued spectrally positive Levy 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 Levy fields recently obtained in (Electron. J. Probab. 25 (2020) 26) and an extension of the Lamperti representation in higher dimension proved in (Ann. Inst. Henri Poincare Probab. Stat. 53 (2017) 1280-1304).