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 n [set membership, variant] {0,...,N - 1}, N [greater-or-equal, slanted] 8 (where g(X-1) [reverse not equivalent] 0). We prove that under monotonicity (or Lipschitz) conditions over the coefficients f, g and [psi], there exists a unique solution {Z1,..., ZN} for this problem and we study its Markov property. The main result that we are able to prove is that the two-dimensional process {(Zn, Zn+1, 1 [less-than-or-equals, slant] n [less-than-or-equals, slant] N - 1} is a reciprocal Markov chain if and only if both the functions f and g are affine.