Stochastic Schrödinger equations as limit of discrete filtering

We consider an open model possessing a Markovian quantum stochastic limit and derive the limit stochastic Schrödinger equations for the wave function conditioned on indirect observations using only the von Neumann projection postulate. We show that the diffusion (Gaussian) situation is universal as a result of the central limit theorem with the quantum jump (Poissonian) situation being an exceptional case. It is shown that, starting from the correponding limiting open systems dynamics, the theory of quantum filtering leads to the same equations, therefore establishing consistency of the quantum stochastic approach for limiting Markovian models.