Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

Abstract

Let X be a (quasi-)Banach space. Let d = (dn) n≥1 be an X-valued sequence of random variables adapted to a filtration (Fn) n≥1 on a probability space (Ω, A, P), define F∞ := σ(Fn : n ≥ 1) and let e = (en) n≥1 be a F∞-conditionally independent sequence on (Ω, A, P) such that L(dn | F n−1 ) = L(en | F∞) for all n ≥ 1 (F 0 = {Ω, ∅}). If there exists a p ∈ (0, ∞) and a constant Dp independent of d and e such that one has, for all n ≥ 1,

A note on maximal estimates for stochastic convolutions

“…However, also in this setting there is some hope that C γ p,Y C X √ p for p large, and by the argument in Theorem 2.4 this would yield exponential tail estimates again. Recently, in [8] it has been proved that C γ p,Y C X p for p large. This yields exponential estimates, but no exponential quadratic estimates as one would expect.…”

The UMD Property for Musielak–Orlicz Spaces

“…However, in this case L Φ (T ) is not a Banach space, but only a quasi-Banach space. Some evidence for the conjecture can be found in [11,Theorem 4.1] and [19, Theorem 1.1] where analogues of Lemma 2.6 can be found (only Φ ∈ ∆ 2 is needed in the proof). Doob's inequality plays a less prominent role for UMD − because of [11,Lemma 2.2].…”

“…Some evidence for the conjecture can be found in [11,Theorem 4.1] and [19, Theorem 1.1] where analogues of Lemma 2.6 can be found (only Φ ∈ ∆ 2 is needed in the proof). Doob's inequality plays a less prominent role for UMD − because of [11,Lemma 2.2]. Similar questions can be asked for the possibly more restrictive "decoupling property" of a quasi-Banach space X introduced in [10,11].…”

“…This property turns out to be p-independent, and it gives a more general class of Banach spaces than the UMD spaces (see [9,10,11,18]). For instance L 1 is a UMD − PW space.…”

On stochastic conservation laws and Malliavin calculus

“…To prove this result we use the Burkholder-Davis-Gundy inequality for real-valued processes. Using Banach space valued versions [12,33], one can derive more general estimates.…”