On the Markov property of a stochastic difference equation
Let \(\{X_ n\), \(0\leq n\leq N-1\}\), \(N>6\), be a discrete time process which satisfies the two-point boundary condition stochastic difference system \[ X_{n+1}= X_ n+ f(X_ n)+ \sigma(X_ n)\xi_ n, \;0\leq n\leq N-1, \qquad F_ 0 X_ 0+ X_ N=F, \tag \(*\) \] where \(\{\xi_ i\), \(0\leq i\leq N-1\}\) are independent variables, \(f\in C^ 2\), \(\sigma\in C^ 3\) with \(f(0)= 0=\sigma(0)\), \(f'(x)\geq -1\), \(\sigma'(x)>0\) for all \(x\) in \((0,\infty)\), the \(\xi_ n\) are positive, and \(F_ 0>0\), \(F>0\). The authors show that \((*)\) has a unique solution and give necessary and sufficient conditions that \((*)\) be a reciprocal Markov chain. [The authors call a sequence of random variables \(\{X_ 0,\dots, X_ M\}\) a reciprocal Markov chain if for every \(m\) and \(n\) with \(0\leq m< n-1<M-1\), the \(\sigma\)-fields \(\sigma(X_{m+1},\dots, X_ n)\) and \(\sigma(X_ 0,\dots, X_ m, X_{n+1},\dots, X_ M)\) are conditionally independent given \(X_ n\) and \(X_ m\)].
two-point boundary condition stochastic difference system
scientific article; zbMATH DE number 691007