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Random Substitution of Time in Strong Markov Processes
V. A. Volkonskii Moscow
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
$\mathrm A$ 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.
Received: 12.03.1958