Stochastic integral representation of bounded quantum martingales in Fock space

Abstract

It is a classical theorem of Kunita and Watanabe [6] that every square integrable martingale adapted to the standard Brownian motion can be uniquely expressed as the stochastic integral of a nonanticipating square integrable Brownian functional with respect to the same Brownian motion. The central aim of this paper is to establish a similar integral represen- tation of a quantum martingale with respect to the annihilation, creation, and gauge processes in the context of quantum stochastic calculus in Fock space as developed in [ 11. In the case of non-Fock quantum Brownian motion such an integral representation was achieved by Hudson and Lindsay [2, 33 with much less difficulty owing to the non-existence of the so called gauge processes. As special cases of our main result we obtain the differentials of Hilbert- Schmidt and unitary martingales [4]. As an application the uniqueness of Fermion martingales in boson Fock space is established. The first author wishes to thank Hudson and Lindsay for several useful conversations on this subject. 2.