Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

By Zdzisław Brzeźniak, J. V. Neerven, Mark Veraar, Lutz Weis. 2008

93 citations recorded by the source corpus.

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/ddcaebdd58f3f83309a0d32040641f6a1add94fb

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. Mark Veraar
2. J. V. Neerven
3. Lutz Weis
4. Arnulf Jentzen
5. Giuseppe Da Prato
6. Sonja G. Cox
7. Petru A. Cioica-Licht
8. Nicolai V. Krylov

In Control and Systems Engineering

1. Mark Veraar
2. Nicolai V. Krylov
3. Giuseppe Da Prato
4. Lutz Weis
5. M. Roeckner
6. J. V. Neerven
7. Sonja G. Cox
8. Arnulf Jentzen

In Finance

1. Mark Veraar
2. Lutz Weis
3. J. V. Neerven
4. Arnulf Jentzen
5. Sonja G. Cox
6. Petru A. Cioica-Licht
7. Nicolai V. Krylov
8. M. Roeckner

Citations in the corpus

• A relatively short proof of Itô’s formula for SPDEs and its applications (2012, background): “See, for instance, [4] and the references therein.”
• A mild Itô formula for SPDEs (2010, background): “4 in Brzeźniak, Van Neerven, Veraar & Weis [9]) give”
• Stochastic integration in quasi-Banach spaces (2006, background): “In order to keep this paper at a reasonable length, the proof of an Itˆo formula is postponed to the paper [ 26 ].”
• Itô’s formula for the Lp-norm of stochastic $${W^{1}_{p}}$$ -valued processes (2008, unknown): “Stochastic partial differential equations, divergence equations, Itô’s formula.”
• Stochastic Integration in Banach Spaces - a Survey (2013, methodology): “SPDEs in Lq((0, τ ]], Lp) spaces.”
• Continuous local martingales and stochastic integration in UMD Banach spaces (2007, background): “The only nontrivial extension of the arguments in [17] is to construct stochastically integrable processes (φn)n≥1 taking values in a finite dimensional subspace of E such that φ = limn→∞ φn in γ(R+, [M ];E) ∩ L(2)(R+, [M ];E) almost surely.”