Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces
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 , we show that the following two-sided inequality holds for all 1 ≤ p < ∞ : Here γ ( [ [ M ] ] t ) is the L 2 -norm of the unique Gaussian measure on X...
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| Published in: | Communications in mathematical physics Vol. 379; no. 2; pp. 417 - 459 |
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| 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
, we show that the following two-sided inequality holds for all
1
≤
p
<
∞
:
Here
γ
(
[
[
M
]
]
t
)
is the
L
2
-norm of the unique Gaussian measure on
X
having
[
[
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 (
⋆
) was proved for UMD Banach functions spaces
X
. We show that for continuous martingales, (
⋆
) holds for all
0
<
p
<
∞
, and that for purely discontinuous martingales the right-hand side of (
⋆
) can be expressed more explicitly in terms of the jumps of
M
. For martingales with independent increments, (
⋆
) is shown to hold more generally in reflexive Banach spaces
X
with finite cotype. In the converse direction, we show that the validity of (
⋆
) 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. |
|---|---|
| AbstractList | In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M0=0, we show that the following two-sided inequality holds for all 1≤p<∞: Here γ([[M]]t) is the L2-norm of the unique Gaussian measure on X having [[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 (⋆) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (⋆) holds for all 0<p<∞, and that for purely discontinuous martingales the right-hand side of (⋆) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (⋆) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (⋆) 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. 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. 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 , we show that the following two-sided inequality holds for all 1 ≤ p < ∞ : Here γ ( [ [ M ] ] t ) is the L 2 -norm of the unique Gaussian measure on X having [ [ 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 ( ⋆ ) was proved for UMD Banach functions spaces X . We show that for continuous martingales, ( ⋆ ) holds for all 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ( ⋆ ) can be expressed more explicitly in terms of the jumps of M . For martingales with independent increments, ( ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ( ⋆ ) 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. |
| Author | Yaroslavtsev, Ivan |
| Author_xml | – sequence: 1 givenname: Ivan orcidid: 0000-0003-1071-6718 surname: Yaroslavtsev fullname: Yaroslavtsev, Ivan email: yaroslavtsev.i.s@yandex.ru organization: Max Planck Institute for Mathematics in the Sciences, Delft Institute of Applied Mathematics, Delft University of Technology |
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| Snippet | 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
, we show... 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 ,... In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M0=0, we show that the... |
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| SubjectTerms | Banach spaces Brownian motion Classical and Quantum Gravitation Complex Systems Covariance Inequalities Integrals Isomorphism Martingales Mathematical and Computational Physics Mathematical Physics Physics Physics and Astronomy Quantum Physics Relativity Theory Theoretical |
| Title | Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces |
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