Semi-Compatible Quantum Finance Models

Abstract

Two observables in a quantum probability space may not be compatible unless they commute. In this note, each basic asset in a market is assumed to commute with the history of trading strategies. A modified quantum finance model, in contrast to non-commuting ones, termed semi-compatible quantum finance model in this note, can be defined with this additional feature. The fundamental asset pricing theorem resumes its original form. It is shown that a self-financing strategy is again self-financing after numeraire change in the Boson Fock space quantum finance model described in Su H., A Quantum Finance Model, Preprint, (2014) and in the Free Brownian motion quantum finance model constructed in Su H., Quantum Finance Models, Preprint, (2015). The proofs are applications of respective Ito formula which dictates the corresponding quantum stochastic calculus. The two very different Ito correction terms which acknowledge the intrinsic distinctions between tensor independence and the free independence in quantum probability lead to individual calculations.