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.

MSC:

 60H99 Stochastic analysis
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References:

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