## Abstract

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.

Original language | English |
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Pages (from-to) | 1988-2018 |

Number of pages | 31 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 55 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Canonical decomposition
- Continuous-time martingales
- Gundy’s decomposition
- Meyer–Yoeurp decomposition
- UMD spaces
- Weak differential subordination
- Weak estimates
- Yoeurp decomposition