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.
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