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The evolution of a continuous time Markov process with a finite number ofstates is usually calculated by the Master equation - a linear differentialequations with a singular generator matrix. We derive a general method forreducing the dimensionality of the Master equation by one by using theprobability normalization constraint, thus obtaining a affine differentialequation with a (non-singular) stable generator matrix. Additionally, thereduced form yields a simple explicit expression for the stationary probabilitydistribution, which is usually derived implicitly. Finally, we discuss theapplication of this method to stochastic differential equations.

An exact reduction of the master equation to a strictly stable system with an explicit expression for the stationary distribution