Existence of densities for multi-type continuous-state branching processes with immigration.

Let X be a multi-type continuous-state branching process with immigration on state space R + d. Denote by g t , t ≥ 0 , the law of X (t). We provide sufficient conditions under which g t has, for each t > 0 , a density with respect to the Lebesgue measure. Such density has, by construction, some Besov regularity. Our approach is based on a discrete integration by parts formula combined with a precise estimate on the error of the one-step Euler approximations of the process. As an auxiliary result, we also provide a criterion for the existence of densities of solutions to a general stochastic equation driven by Brownian motions and Poisson random measures, whose coefficients are Hölder continuous and might be unbounded. [ABSTRACT FROM AUTHOR]