On a stochastic delay difference equation with boundary conditions and its Markov property

In the present paper we consider the one-dimensional stochastic delay difference equation with boundary condition [GRAPHICS] n is an element of {0,...,N-1}, N greater than or equal to 8 (where g(X(-1))=0) We prove that under monotonicity (or Lipschitz) conditions over the coefficients f,g and psi, there exists a unique solution {Z(1),...,Z(N)} for this problem and we study its Markov property. The main result that we are able to prove is that the two-dimensional process {(Z(n),Z(n+1)), 1 less than or equal to n less than or equal to N-1} is a reciprocal Markov chain if and only if both the functions f and g are affine.