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+F(Xn)+g(Xn−1)+ξn,X0=ψ(XN), ,…,N − 1}, N ⩾ 8 (where g(X−1) ≡ 0). We prove that under monotonicity (or Lipschitz) conditions over the coefficients f, g and ψ, 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 ⩽ n ⩽ N − 1} is a reciprocal Markov chain if and only if both the functions f and g are affine.