Stop times in Fock space quantum probability

Abstract

We review Fock space based quantum probability and in particular, the theory of stop times based on it. In the Fock space , a stop time may be defined as a positive self-adjoint operator whose spectral resolution is adapted to the natural filtration based on the splittings

Then if K is one of the basic quantum martingales (creation, preservation or annihilation) the corresponding process

beginning anew at time T can be defined unambiguosly using functional calculus for the self-adjoint operator T with an operator-valued integrand

since the integrand commutes with the integrator. An isometric operator , the forward shift through T, is defined by a similar spectral integral of sure forward shifts and conjugates with . An optional stopping theorem is formulated and proved using similar operator valued spectral integrals.

Keywords:

• Quantum probability
• Stop times
• Strong Markov
• Optional stopping

Mathematics Subject Classification::

• 60G40
• 81S25

Notes

All Hilbert spaces are complex; inner products are linear in the second entry.

In this paper, we do not consider stop times which can take the value ∞.

The three possibilities are known, at least to quantum probabilists who are also violinists, as left, right and double stopping.