On a stochastic delay difference equation with boundary conditions and its Markov property. (English) Zbl 0846.60063

Summary: We consider the one-dimensional stochastic delay difference equation with boundary condition \[ \begin{cases} X_{n + 1} = X_n + f(X_n) + g(X_{n - 1}) + \xi_n, \\ X_0 = \psi (X_N), \end{cases} \] \(n \in \{0, \dots, N - 1\}\), \(N \geq 8\) (where \(g(X_{-1}) \equiv 0)\). We prove that under monotonicity (or Lipschitz) conditions over the coefficients \(f,g\) and \(\psi\), there exists a unique solution \(\{Z_1, \dots, Z_N\}\) for this problem and we study its Markov property. The main result that we are able to prove is that the two-dimensional process \(\{(Z_n, Z_{n + 1})\), \(1 \leq n \leq N - 1\}\) is a reciprocal Markov chain if and only if both the functions \(f\) and \(g\) are affine.


60H99 Stochastic analysis
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