Existence of densities for multi-type CBI processes

Let X be a multi-type continuous-state branching process with immigration (CBI process) on state space $\mathbb{R}^d$. Denote by $g_t$, $t \geq 0$, 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 anisotropic Besov regularity. Our approach neither relies on the use of Malliavin calculus nor on the study of corresponding Laplace transform.