Continuity of filtrations of sigma algebras (Q1096953)

The relationship between the continuity of sample paths and the continuity of the canonical filtration of a stochastic process is explored. The authors give two counterexamples to show that no such relationship exists; a process can have \(C^{\infty}\) sample paths and a discontinuous filtration or it can have a continuous filtration and arbitrarily irregular sample paths. After this, a characterization of all processes with continuous filtration is obtained and then this characterization is used to show that in a separable probability space a filtration can only have countably many points of discontinuity.