Stationary measure of random process of drift with reflecting barriers in semi-Markov environment. (Ukrainian, English) Zbl 1150.60042

Pogoruj, A. O.

Teor. Jmovirn. Mat. Stat. 74, 108-115 (2006); translation in Theory Probab. Math. Stat. 74, 125-132 (2007).

The process of drift in semi-Markov environment is described by equation \({dv(t)\over dt}=C(k(t),v(t))\), where \(k(t)\) is semi-Markov process with phase space \(G=X\cup Y\), \(X=\{x_1,\dots,x_{n}\}\), \(Y=\{y_1,\dots,y_{m}\}\). Let \(V_0,V_1, a_{i}, b_{j}\in \mathbb{R}\), \(V_0<V_1\), \(a_{i}>0, b_{j}>0\), \(i=1,\dots,n\), \(j=1,\dots,m\), and for all \(x_{i}\in X, i=1,\dots,n\): \(C(x_{i},v)=\begin{cases} -a_{i},& V_0<v\leq V_1 ,\\ 0,& v=V_0,\end{cases}\) for all \(y_{j}\in X, j=1,\dots,m\): \(C(y_{j},v)=\begin{cases} b_{j},& V_0\leq v< V_1,\\ 0,& v=V_1.\end{cases} \) The author studies the stationary measure of the process \(\xi(t)=(\tau(t), k(t), v(t))\), where \(\tau(t)=t-\sup\{u\leq t:k(t)\neq k(u)\}\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60K15

Markov renewal processes, semi-Markov processes

Keywords:

stationary measure; random process of drift; reflecting barriers; semi-Markov environment