Multiple Q-Adapted Integrals and Ito Formula of Noncommutative Stochastic Calculus in Fock Space

TL;DR

This paper develops a generalized Ito formula for noncommutative quantum stochastic calculus in Fock space, focusing on multiple Q-adapted integrals and their continuity properties.

Contribution

It introduces a new differential Q-adapted Ito formula that extends the classical Ito product rule to noncommutative quantum stochastic integrals.

Findings

Established continuity of multiple Q-adapted integrals

Formulated a generalized Ito product formula in noncommutative setting

Analyzed the algebra of nonadapted quantum stochastic processes

Abstract

We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced by Belavkin in 1991. The differential Q-adapted formula generalizing Ito product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Ito formula.