Quantum stopping times and quasi-left continuity

Abstract

We study the general properties of quantum stopping times on Hilbert spaces equipped with a filtration. We define and investigate notions such as the spaces of anterior events, the spaces of strictly anterior events and above all we define the property S< T for two stopping times together with the notion of predictable quantum stopping times. It is well-known that the natural filtration of any normal martingale with the predictable representation property is quasi-left continuous; with the help of our new notions we prove that this property is actually an intrinsic property of the symmetric Fock space Φ over L 2( R +) . We also apply these definitions to the case of a non commutative stochastic base. We show, in this context, that the fermionic Fock space over L 2( R +) , the quasi-free boson and fermion spaces are also quasi-left continuous.