Abstract 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 Holder continuous and might be unbounded.

中文翻译：

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具有迁移的多类型连续状态分支过程的密度存在性

摘要 设 X 是状态空间 R + d 上具有迁移的多类型连续状态分支过程。用gt表示，t≥0，X(t)定律。我们提供了充分条件，在该条件下，对于每个 t > 0，gt 具有关于 Lebesgue 测度的密度。这种密度在构造上具有某种 Besov 规律。我们的方法基于分部公式的离散积分，并结合对过程的一步欧拉近似误差的精确估计。作为辅助结果，我们还提供了一个由布朗运动和泊松随机测度驱动的一般随机方程解的密度存在性的标准，其系数是霍尔德连续的并且可能是无界的。