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Existence of densities for multi-type continuous-state branching processes with immigration. (English) Zbl 1453.60040

Summary: Let \(X\) be a multi-type continuous-state branching process with immigration 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 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.

MSC:

60E07 Infinitely divisible distributions; stable distributions
60G30 Continuity and singularity of induced measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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