The optional stopping theorem for quantum martingales

Abstract

In classical probability theory, a random time T is a stopping time in a filtration ( F t ) t ⩾ 0 if and only if the optional sampling holds at T for all bounded martingales. Furthermore, if a process ( X t ) t ⩾ 0 is progressively measurable with respect to ( F t ) t ⩾ 0 , then X T is F T -measurable. Unfortunately, this is not the case in noncommutative probability with the definition of stopped process used until now. It is shown in this article that we can define the stopping of noncommutative processes in Fock space in such a way that all the bounded martingales can be stopped at any stopping time T, are adapted to the filtration of the past before T and satisfy the optional stopping theorem.