On the Markov property of a stochastic difference equation

In the present paper we study the one-dimensional stochastic difference equation Xn+1 = Xn + f(Xn) + [sigma](Xn)[xi]n, n [epsilon] {0, ..., N - 1}, N \s#62; 6, with linear boundary conditions at the endpoints. We present an existence and uniqueness result and study the Markov property of the solution. We are able to prove that the solution is a reciprocal Markov chain if and only if the functions f(x) and [sigma](x) are both polynomial out of a "small" interval, whose length depends on f and the boundary condition.