The covariation for Banach space valued processes and applications
This article focuses on a recent concept of covariation for processes taking values in a separable Banach space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$B$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>B</mi> </math> </EquationSource> </InlineEquation> and a corresponding quadratic variation. The latter is more general than the classical one of Métivier and Pellaumail. Those notions are associated with some subspace <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\chi $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">χ</mi> </math> </EquationSource> </InlineEquation> of the dual of the projective tensor product of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$B$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>B</mi> </math> </EquationSource> </InlineEquation> with itself. We also introduce the notion of a convolution type process, which is a natural generalization of the Itô process and the concept of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$\bar{\nu }_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mover accent="true"> <mrow> <mi mathvariant="italic">ν</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> <mn>0</mn> </msub> </math> </EquationSource> </InlineEquation>-semimartingale, which is a natural extension of the classical notion of semimartingale. The framework is the stochastic calculus via regularization in Banach spaces. Two main applications are mentioned: one related to Clark–Ocone formula for finite quadratic variation processes; the second one concerns the probabilistic representation of a Hilbert valued partial differential equation of Kolmogorov type. Copyright Springer-Verlag Berlin Heidelberg 2014
