Quantum stochastic integrals as operators

TL;DR

This paper develops a framework for quantum stochastic integrals where the integrator is a martingale within a von Neumann algebra, extending classical stochastic calculus into the quantum domain.

Contribution

It introduces a method to construct quantum stochastic integrals with integrators as martingales in von Neumann algebras, including the finite algebra case with $L^2$--martingales.

Findings

Constructed quantum stochastic integrals for martingales in von Neumann algebras.

Extended the theory to finite algebras with $L^2$--martingale integrators.

Showed that the integrals are $L^2$--martingales in the finite algebra case.

Abstract

We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand -- a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an $L^2$--martingale in which case the integrals are $L^2$--martingales too.