  We show that the canonical decomposition (comprising both the Meyer-Yoeurp
and the Yoeurp decompositions) of a general $X$-valued local martingale is
possible if and only if $X$ has the UMD property. More precisely, $X$ is a UMD
Banach space if and only if for any $X$-valued local martingale $M$ there exist
a continuous local martingale $M^c$, a purely discontinuous quasi-left
continuous local martingale $M^q$, and a purely discontinuous local martingale
$M^a$ with accessible jumps such that $M = M^c + M^q + M^a$. The corresponding
weak $L^1$-estimates are provided. Important tools used in the proof are a new
version of Gundy's decomposition of continuous-time martingales and weak
$L^1$-bounds for a certain class of vector-valued continuous-time martingale
transforms.
