A counterexample on two-parameter Markov processes (Q909350)

The author analyzes the relation between different versions of the Markov property for processes in the plane and gives some counterexamples to \textit{H. Korezlioglu} and \textit{P. Lefort}, C. R. Acad. Sci. Paris, Ser. A 290, 555-558 (1980; Zbl 0435.60048). Let \(X=\{x_ z\), \(z\in R^ 2_+\}\) be a stochastic process, \(x_ z=c\) for \(z\in R^ 2\setminus R^ 2_+\). We say that X is large-past Markov if for any \(z\leq z'\), \(z,z'\in R^ 2_+:\) \[ E(f(x_{z'}| \quad {\mathcal F}^*_ z)=E(f(x_{z'})| \quad x_ z,\quad x_{z\circ z'},x_{z'}); \] X is *-Markov if this relation holds for any \(z,z'\in R^ 2.\) The paper states that X is *-Markov if and only if X is large-past Markov and \(\{x_{st}\), \({\mathcal F}^ 1_ s\), \(s\geq 0\}\), \(\{x_{st}\), \({\mathcal F}^ 2_ t\), \(t\geq 0\}\) are one-parameter Markov processes.