Random Substitution of Time in Strong Markov Processes

Abstract

The terminology and symbols are as in [7] and [1].Let $x(t,\omega )$ be a homogeneous strong Markov process, and $\tau _t (\omega )$ be a random function not decreasing for increasing t. The process $y_t = x(\tau _t (\omega ),\omega )$ is called a process obtained from $x_t (\omega )$ by means of a random substitution of time $\tau _t $.The conditions sufficient for the process $y_t $ to be a Markov or a strong Markov process are formulated (Theorems 1 and 2).In [1] it is shown that the infinitesimal operator A of a Feller strong Markov process continuous on the right is a contraction of a certain operator $\mathfrak{A}$, which is called the extended operator. It is shown that if $x_t $ and $x(\tau _t )$ are Feller processes continuous on the right and $\tau _t $ is determined by equation (2), where $\varphi (x) > 0$, and continuous, then their extended operator is $\mathfrak{A}$, where $\mathfrak{A}$ satisfies the equation $t = \varphi (x)\mathfrak{A}$ (Theorem 3).In Theorem 4 and in its corollary it is ...