Continuous local martingales and stochastic integration in UMD Banach spaces

By Mark Veraar. 2007

213 citations recorded by the source corpus.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7a4bca77cfdaee73a3e66e5d6f72b9d2c5719ecc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. J. V. Neerven
2. Mark Veraar
3. Lutz Weis
4. Lucio Galeati
5. Arnulf Jentzen
6. Sonja G. Cox
7. Petru A. Cioica-Licht
8. Massimiliano Gubinelli

In Finance

1. J. V. Neerven
2. Mark Veraar
3. Lutz Weis
4. Lucio Galeati
5. Giuseppe Da Prato
6. Massimiliano Gubinelli
7. Sonja G. Cox
8. Petru A. Cioica-Licht

In Mathematical Physics

1. J. V. Neerven
2. Mark Veraar
3. Lutz Weis
4. Lucio Galeati
5. Sonja G. Cox
6. Petru A. Cioica-Licht
7. Massimiliano Gubinelli
8. Zdzisław Brzeźniak

Citations in the corpus

• Stochastic integration in quasi-Banach spaces (2006, methodology): “Regarding stochastic integrands, as already mentioned above, our approach is similar in spirit to the stochastic integration theory in Banach spaces with the UMD property that was developed in [71] (a survey can be found in [73]).”
• Uniformly gamma-radonifying families of operators and and the stochastic Weiss conjecture (2006, unknown): “In a parallel development, γ–radonifying norms have been used recently to extend the theory of stochastic integration to the Banach space setting, first for operatorvalued functions taking values in arbitrary Banach spaces [36, 37] and subsequently for operator-valued processes taking values in UMD ”
• Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation (2008, methodology): “Using the theory of stochastic integration in UMD spaces developed recently in [23], an Itô formula is proved which is subsequently applied to the duality mapping defined on the UMD space E × E, (x, x) 7→ 〈x, x〉.”
• Noiseless regularisation by noise (2020, background): “In the above statement, γ(L(2)(R+), X) stands for the p-th γ-Radonifying norm; we omit the precise definition, which can be found in [49], [50].”
• Stochastic Integration in Banach Spaces - a Survey (2013, methodology): “Earlier versions of this result, as well as related characterisations, can be found in [76, 77].”
• The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates (2016, unknown): “ide of (2.3) in Lq(0;T;Lp(Rd))-spaces for q;p > 2:This can be done by the theory of stochastic integration in martingale type 2 spaces, see [Brz97] and the references therein or in UMD spaces, see [vNVW07]. The tool to estimate the stochastic convolution (2.3) is the following result due to Brzez”
• General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions (2012, background): “[21, 22]) does not fit the present case because, if a Banach space is UMD, then it is reflexive.”
• A mild Itô formula for SPDEs (2010, background): “ues in UMD Banach spaces (see Theorem 2.4 in Brze´zniak, Van Neerven, Veraar & Weis [9]). Details on the stochastic integration in UMD Banach spaces can be found in Van Neerven, Veraar & Weis [102, 103] and in the references therein. Another possible generalization is to consider more genera”