  We prove that a single-jump quantum stochastic unitary evolution is
equivalent to a Dirac boundary value problem on the half line in one extra
dimension. It is shown that this exactly solvable model can be obtained from a
Schroedinger boundary value problem for a positive relativistic Hamiltonian in
the half-line as the inductive ultrarelativistic limit, correspondent to the
input flow of Dirac particles with asymptotically infinite momenta. Thus the
problem of stochastic approximation is reduced to the to the quantum-mechanical
boundary value problem in the extra dimension. The question of microscopic time
reversibility is also studied for this paper.
