Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces

Abstract

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 $$M_0=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> , we show that the following two-sided inequality holds for all $$1\le p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> : Here $$ \gamma ([\![M]\!]_t) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>γ</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mspace/><mml:mrow><mml:mo>[</mml:mo><mml:mi>M</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mspace/><mml:mo>]</mml:mo></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math> is the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> -norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mspace/><mml:mrow><mml:mo>[</mml:mo><mml:mi>M</mml:mi><mml:mo>]</mml:mo></mml:mrow><mml:mspace/><mml:mo>]</mml:mo></mml:mrow><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mo>⟨</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>⟩</mml:mo></mml:mrow><mml:mo>,</mml:mo><mml:mrow><mml:mo>⟨</mml:mo><mml:mi>M</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mo>∗</mml:mo></mml:msup><mml:mo>⟩</mml:mo></mml:mrow><mml:mo>]</mml:mo></mml:mrow><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math> as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ) was proved for UMD Banach functions spaces X . We show that for continuous martingales, ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ) holds for all $$0&lt;p&lt;\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo>&lt;</mml:mo><mml:mi>p</mml:mi><mml:mo>&lt;</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:math> , and that for purely discontinuous martingales the right-hand side of ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ) can be expressed more explicitly in terms of the jumps of M . For martingales with independent increments, ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ( $$\star $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>⋆</mml:mo></mml:math> ) 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.

Topics & Concepts