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2 papers
Short Communications
«Splitting times» for random fields
I. V. Evstigneev,
A. I. Ovseevič Moscow
Abstract:
There are various results concerning the strong Markov property for one-dimensional processes. The essence of these results can be expressed as follows. For a «good» Markov process
$x_t$ and a special class of random times
$\tau$ (the so called optional times): a) the behaviour of the process
$x_t$ before
$\tau$ and its behaviour after
$\tau$ are conditionally independent given
$\tau$,
$x_{\tau}$; b) the forecast of the process' behaviour after
$\tau$ based on known
$\tau$,
$x_{\tau}$ is quite the same as if
$\tau$ be non-random (this forecast is determined by the transition function
$p(\tau,x_{\tau},s,\Gamma)$).
Williams and Jacobsen introduced a class of random times
$\tau$, the so called splitting times, for which property a) only is valid. In this paper, the concept of splitting time is generalized for random fields.
A random field is defined as a system of
$\sigma$-algebras
$\{\mathscr F_V\}$,
$V$ being a closed subset of a finite-dimensional Euclidean space
$X$. The Markov property means that
$\mathscr F_V$ and
$\mathscr F_W$ are conditionally independent given
$\mathscr F_{V\bigcap W}$ provided
$V\bigcup W=X$. A random time is a pair of random closed sets
$V$ and
$W$ such that
$V\bigcup W=X$. Given
$\tau=(V,W)$, we introduce the
$\sigma$-algebras of the «past»
$\mathscr F_V$, the «future»
$\mathscr F_W$ and the «present»
$\mathscr F_{V\bigcap W}$. A random time
$\tau$ is called a splitting time if the past and the future are conditionally independent given the present. We give necessary and sufficient conditions for a random time with countably many values to be splitting and find sufficient conditions in the case of uncountably many values.
Received: 23.09.1976