Suppose a brownian motion $W$ and a poisson random measure $\mu$ are defined on the same filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t), P)$, where both $W$ and $\mu$ are $\mathcal{F}_t)$ adapted. Then does it have to be the case that $W$ is independent of $\mu$? Thanks!
2026-04-08 18:08:04.1775671684
does brownian motion and poisson random measure have to be independent?
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Yes, they are independent; for a proof see e.g.
N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes, Theorem II.6.3.
Note that a similar question (for the case of Poisson processes, not Poisson random measures) has been discussed on mathoverflow.