Quantum Ω-Semimartingales and Stochastic Evolutions

✍ Scribed by Alexander C.R Belton

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 193 KB
• Volume: 187
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3809

✦ Synopsis

We explore W-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, W-adapted processes are themselves bounded and W-adapted, a fact that may be deduced from the Bismut-Clark-Ocone formula of Malliavin calculus. An algebra analogous to Attal's class S of regular quantum semimartingales is defined, and product and functional Ito ˆformulae are given. We consider quantum stochastic differential equations with bounded, W-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.