## Abstract

In this paper, we consider Meyer–Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞), any X-valued L^{p}-martingale M has a unique decomposition M = M^{d} + M^{c} such that M^{d} is a purely discontinuous martingale, M^{c} is a continuous martingale, M_{0} ^{c} = 0 and EM_{∞} ^{d p} + EM_{∞} ^{c p} ≤ c_{p,X}EM_{∞} ^{p}. An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application, we show that X is a UMD Banach space if and only if for any fixed p ∈ (1, ∞) and for all X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate EN_{∞} ^{p} ≤ C_{p,X}EM_{∞} ^{p}

Original language | English |
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Pages (from-to) | 1659-1689 |

Number of pages | 31 |

Journal | Bernoulli |

Volume | 25 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Accessible jumps
- Brownian representation
- Burkholder function
- Canonical decomposition of martingales
- Continuous martingales
- Differential subordination
- Meyer–Yoeurp decomposition
- Purely discontinuous martingales
- Quasi-left continuous
- Stochastic integration
- UMD Banach spaces
- Weak differential subordination
- Yoeurp decomposition