Itô Formula for Processes Taking Values in Intersection of Finitely Many Banach Spaces

Abstract

Motivated by applications to SPDEs we extend the Itô formula for the square of the norm of a semimartingale $y(t)$ from Gyöngy 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 Gyöngy and Krylov (Stochastics 6(3):153-173, 1982) to the semimartingale above.