A non-commutative martingale representation theorem for non-Fock quantum Brownian motion

A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

“…To establish the Lévy-Cielsieski construction for quantum Brownian motion of variance σ 2 > 1, we follow [11] and work in Γ(l 2 (Z + )) ⊗ Γ(l 2 (Z + )), where · here denotes duality. In place of each ψ(f ) use ψ(f ) ⊗ ψ(f ) and…”

Quantum stochastic integrals as belated integrals

“…Equip the linear space X with the topology induced by the family of seminorms F^iS'o II^X y )V'(/)ll 2 'fr} 1/2 indexed b y / e 2) and f e R + . It was shown in [10] that the elements of 3if can be integrated with respect to the Fock creation, preservation and annihilation processes, thus defining quantum stochastic integrals constructed in [9,12] are belated integrals (one uses the explicit representation on 36 <8> VC as given, for example, in [8]). However, it was shown in [6] that the integrals of [4] are belated integrals, and in [13] it was shown that the integrals of [9] are the same as a class of those constructed in [4].…”

“…Quantum stochastic integrals have been constructed in various contexts [2,3,4,5,9] by adapting the construction of the classical L 2 -It6-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands.…”

Quantum stochastic calculus on interacting Fock spaces: semimartingale estimates and stochastic integral

“…After these pioneering works, a great number of papers was devoted to develop a theory in non Boson cases (see e.g. [10] for the Fermion case, [15] for universal invariant case, [24] for free, [18] for general quasi-free, [12] for Boolean, [23] for full Fock module). Accardi, Fagnola and Quaegebeur in [3] reached a double result: on the one hand developing a theory independent of the particular representation chosen (as in the classical case) and on the other hand including all the quantum stochastic calculi already appeared (boson and fermion) into a unifying picture.…”