It\xf4 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 ∑i=1m∫(0,t]vi∗(s)dA(s)+h(t)=:y(t)∈VdA×P-a.e.,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\sum _{i=1}^m \\int _{(0,t]} v_i^{*}(s)\\,dA(s) + h(t)=:y(t)\\in V \\quad dA\\times {\\mathbb {P}}\\text {-a.e.}, \\end{aligned}$$\\end{document}where A is an increasing right-continuous adapted process, v_i^{*} is a progressively measurable process with values in V_i^{*}, 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^{*}, and V:=V_1cap V_2 cap cdots cap V_m is continuously and densely embedded in H. The formula is proved under the condition that Vert yVert _{V_i}^{p_i} and Vert v_i^*Vert _{V_i^*}^{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.