Stochastic realization problems are presented for a tuple of Gaussian random variables, for a tuple of σ -algebras, for a σ -algebra family, and for a finite stochastic system. The solution of the weak and of the strong stochastic realization of a tuple of Gaussian random variables is provided. The main theoretical contribution is the description of the strong stochastic realization of a tuple of σ -algebras. This is followed by stochastic realization of a family of σ -algebras. Finally the stochastic realization problem for finite-valued output processes is discussed.