On the relative entropy of discrete-time Markov processes with given end-point densities

Given a Markov process x(k) defined over a finite interval I=[0,N], I/spl sub/Z we construct a process x*(k) with the same initial density as x, but a different end-point density, which minimizes the relative entropy of x and x*. It is shown that x* is a Markov process in the same reciprocal class as x. In the Gaussian case, the minimum relative entropy problem is related to a minimum energy LQG optimal control problem.