Random Substitution of Time in Strong Markov Processes

V. A. Volkonskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 3 (1958) no. 3, pp. 332-350

Résumé

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 $\mathrm A$ 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 shown that a one-dimensional homogeneous regular continuous strong Markov process may be obtained from a Wiener process by means of a random substitution of time and a monotone transformation of the segment.