TY - JOUR
T1 - Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces
AU - Yaroslavtsev, Ivan
PY - 2020/10/1
Y1 - 2020/10/1
N2 - 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 , we show that the following two-sided inequality holds for all 1 ≤ p' ∞: [Figure not available: see fulltext.] 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.
AB - 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 , we show that the following two-sided inequality holds for all 1 ≤ p' ∞: [Figure not available: see fulltext.] 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.
UR - http://www.scopus.com/inward/record.url?scp=85091104583&partnerID=8YFLogxK
U2 - 10.1007/s00220-020-03845-7
DO - 10.1007/s00220-020-03845-7
M3 - Article
AN - SCOPUS:85091104583
SN - 0010-3616
VL - 379
SP - 417
EP - 459
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 2
ER -