Abstract
In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that $$M_0=0$$
M
0
=
0
, we show that the following two-sided inequality holds for all $$1\le p<\infty $$
1
≤
p
<
∞
: Here $$ \gamma ([\![M]\!]_t) $$
γ
(
[
[
M
]
]
t
)
is the $$L^2$$
L
2
-norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$
[
[
M
]
]
t
(
x
∗
,
y
∗
)
:
=
[
⟨
M
,
x
∗
⟩
,
⟨
M
,
y
∗
⟩
]
t
as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ($$\star $$
⋆
) was proved for UMD Banach functions spaces X. We show that for continuous martingales, ($$\star $$
⋆
) holds for all $$0<p<\infty $$
0
<
p
<
∞
, and that for purely discontinuous martingales the right-hand side of ($$\star $$
⋆
) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, ($$\star $$
⋆
) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ($$\star $$
⋆
) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.
Inverse Problems via the “Generalized Collage Theorem” for Vector-Valued Lax-Milgram-Based Variational Problems