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.

Highlights

  • Itô formula for the square of the norm is an essential tool in the study of stochastic evolution equations of the type dv(t) = A(t, v(t)) dt + Bk(t, v(t)) d W k(t), (1.1)

  • K where (W k)∞ k=1 is a sequence of independent Wiener processes, and A(t, ·) and Bk (t, ·) are operators on a separable real Banach space V, with values in a Banach space V and a Hilbert space H respectively, such that V → H →

  • A proof, not bound to the theory of SPDEs, was given in Krylov and Rozovskii [12], and this stochastic energy equality was generalised in Gyöngy and Krylov [6] to V ∗-valued semimartingales y of the form y(t) =

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Summary

Introduction

A proof, not bound to the theory of SPDEs, was given in Krylov and Rozovskii [12], and this stochastic energy equality was generalised in Gyöngy and Krylov [6] to V ∗-valued semimartingales y of the form y(t) =. In the present paper we are interested in stochastic energy equalities which can be applied to SPDEs (1.4) when A is of the form A = A1 + A2 + · · · + Am and the operators Ai have different analytic and growth properties. This means, Ai (t, ·) : Vi → Vi i = 1, 2, . For general theory of SPDEs in the variational setting we refer the reader to Krylov and Rozovskii [12], Prévôt and Röckner [15] and Rozovskii [16]

Main results
Preliminaries
Proof of the main result
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