It\^o Formula for Processes Taking Values in Intersection of Finitely Many Banach Spaces

TL;DR

This paper extends the Itô formula to processes in the intersection of multiple Banach spaces, accommodating applications to stochastic partial differential equations and broadening the class of semimartingales for which the formula applies.

Contribution

It introduces a generalized Itô formula for processes in the intersection of finitely many Banach spaces, relaxing previous integrability conditions for broader applicability.

Findings

Extended Itô formula for Banach space intersections

Weaker integrability conditions than previous results

Applicability to stochastic PDEs with complex structures

Abstract

Motivated by applications to SPDEs we extend the It\^o formula for the square of the norm of a semimartingale $y(t)$ from Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the case \begin{equation*} \sum_{i=1}^m \int_{(0,t]} v_i^{\ast}(s)\,dA(s) + h(t)=:y(t)\in V \quad \text{$dA\times \mathbb{P}$-a.e.}, \end{equation*} where $A$ is an increasing right-continuous adapted process, $v_i^{\ast}$ is a progressively measurable process with values in $V_i^{\ast}$, the dual of a Banach space $V_i$, $h$ is a cadlag martingale with values in a Hilbert space $H$, identified with its dual $H^{\ast}$, and $V:=V_1\cap V_2 \cap \ldots \cap V_m$ is continuously and densely embedded in $H$. The formula is proved under the condition that $\|y\|_{V_i}^{p_i}$ and $\|v_i^\ast\|_{V_i^\ast}^{q_i}$ are almost surely locally integrable with respect to $dA$ for some conjugate exponents $p_i, q_i$. This condition is essentially weaker than the one which would arise in application of the results in Gy\"ongy and Krylov (Stochastics 6(3):153-173, 1982) to the semimartingale above.