An algebra of non-commutative bounded semimartingales: Square and angle quantum brackets (Q1340814)

The author shows that the space of adapted processes of bounded operators on a Fock space forms (under certain regularity condition) a \(*\)-algebra of quantum semimartingales. Moreover, characterization of this algebra in terms of the regularity of semimartingales with respect to some Radon measures is given (extension of the quantum martingal representation theorem of Paratharasy and Sinha). On using the algebraic structure of semimartingales and the quantum Itô formula the non-commutative stochastic calculus is developed. Among other results a non-commutative Itô formula for polynomials of non-commutative semimartingales and intrinsic definition of square brackets are given.