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Quantum Ω-semimartingales and stochastic evolutions

Research output: Contribution to journalJournal article

Published

Journal publication date1/12/2001
JournalJournal of Functional Analysis
Journal number1
Volume187
Number of pages16
Pages94-109
Original languageEnglish

Abstract

We explore Ω-adaptedness, a variant of the usual notion of adaptedness found in stochastic calculus. It is shown that the (non-adapted) quantum stochastic integrals of bounded, Ω-adapted processes are themselves bounded and Ω-adapted, a fact that may be deduced from the Bismut–Clark–Ocone formula of Malliavin calculus. An algebra analogous to Attal's class of regular quantum semimartingales is defined, and product and functional Itô formulae are given. We consider quantum stochastic differential equations with bounded, Ω-adapted coefficients that are time dependent and act on the whole Fock space. Solutions to such equations may be used to dilate quantum dynamical semigroups in a manner that generalises, and gives new insight into, that of R. Alicki and M. Fannes (1987, Comm. Math. Phys.108, 353–361); their unitarity condition is seen to be the usual condition of R. L. Hudson and K. R. Parthasarathy (1984, Comm. Math. Phys93, 301–323).