I want to point attention to the following paper

H. P. Breuer & F. Petruccione,

Stochastic dynamics of open quantum systems: Derivation of the

differential Chapman-Kolmogorov equation,

Physical Review E51, 4041-4054 (1995).

The dynamics of a large quantum system, consisting of an observed system and a detector observing it, is treated as a classical dynamical system for the state vector with stochastic initial conditions, and reduced to a classical stochastic equation in the projective Hilbert space of the observed system, using standard assumptions from classical statistical mechanics only. In other words, the derivation is done in the same way as one would proceed in statistical mechanics for any other classical dynamical system.

Since only classical probabilities are used it is impossible for quantum mechanical collapse to enter the argument. But at the end one gets a piecewise deterministic stochastic process (PDP) for the reduced state vector. Only after everything has been done, the PDP is interpreted in terms of quantum jumps.

This proves that collapse in a single observed system (in the modern POVM version of the von Neumann postulates for quantum dynamics) is derivable from the unitary dynamics of a bigger system under the standard assumptions that go into the traditional derivations in classical statistical mechanics.