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Continuity Conditions for Stochastic Processes
L. V. Seregin Moscow
Abstract:
Let
$x_t$,
$0\leq t\leq c<\infty$, be a separable stochastic process in the metric space
$X$. The main purpose of this paper is to derive conditions under which almost all sample functions of the process
$x_t$ are continuous. We designate by
$\rho(x,y)$ the distance between the points
$x,y\in X$.
Let
$\mathbf P(\dots)$ be a Markov transition function, satisfying for each
$\varepsilon>0$ $$\mathop{\sup}\limits_{x,s,t}{\mathbf P}\left({s,x,t,V_\varepsilon (x)}\right)=o(1),\quad h\downarrow 0,$$
where
$x\in X$;
$s,t\in[0,c],0 <t-s\leq h$ and $V_\varepsilon(x)=\{{y:\rho(x,y)\geq\varepsilon}\}$. Then almost all sample functions of the Markov process
$x_t$ are continuous if and only if for each
$\varepsilon>0$ $$\int_0^{c-h}
\mathbf P\{\rho\left(x_t,x_{t+h}\right)>\varepsilon\}\,dt=o(h),\quad h\downarrow 0.$$
Almost all sample functions of a martingale (semi-martingale)
$x_t$ are continuous if and only if for
$h\downarrow 0$ $$\int_0^{c-h}{\mathbf P\left\{{x_t<a,x_{t+h}>b}\right\}\,dt=o(h),}$$
$$\int_0^{c-h}{\mathbf P\left
\{{x_t>b,x_{t+h}< a}\right\}\,dt=o(h)}$$
for each
$a$ and
$b$,
$a<b$.
Received: 29.03.1959