Quantum Ω-semimartingales and stochastic evolutions

We explore Omega -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, Omega -adapted processes are themselves bounded and Omega -adapted, a fact that may be deduced from the Bismut-Clark-Ocone formula of Malliavin calculus. An algebra analogous to Attal's class T of regular quantum semimartingales is defined, and product and functional Ito formulae are given. We consider quantum stochastic differential equations with bounded, Omega -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, Conan. Math. Phys 93, 301-323). (C) 2001 Academic Press.