Continuity Conditions for Stochastic Processes

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 $b$, $a<b$.