Quantum stopping times stochastic integral in the interacting Fock space

摘要

Following the ideas of Hudson [J. Funct. Anal. 34(2), 266-281 (1979)] and Parthasarathy and Sinha [Probab. Theory Relat. Fields 73, 317-349 (1987)], we define a quantum stopping time (QST, for short) tau in the interacting Fock space (IFS, for short), Gamma, over L-2 (R+), which is actually a spectral measure in [0,infinity] such that tau ([0,t]) is an adapted process. Motivated by Parthasarathy and Sinha [Probab. Theory Relat. Fields 73, 317-349 (1987)] and Applebaum [J. Funct. Anal. 65, 273-291 (1986)], we also develop a corresponding quantum stopping time stochastic integral (QSTSI, for abbreviations) on the IFS over a subspace of L-2 (R+) equipped with a filtration. As an application, such integral provides a useful tool for proving that G admits a strong factorisation, i.e., Gamma = Gamma(tau]) circle times Gamma([tau) , where Gamma(]tau) and Gamma([tau) stand for the part "before tau" and the part "after tau," respectively. Additionally, this integral also gives rise to a natural composition operation among QST to make the space of all QSTs a semigroup. (C) 2015 AIP Publishing LLC.