$|X_t - X_s|$ notation in stochastic processes

Question

Let $X_t$ be a real stochastic process. I am new to stochastic processes and in some proofs I see inequalities such as: $$|X_t - X_s| \leq C_1 \quad \forall |t-s| \leq C_2 \tag{1}$$ but $X$ is a function of $t$ and $\omega$, i.e. $X_t(\omega)$ so I am not sure what is meant by (1). Does this mean (1) holds uniform in $\omega$? Is this standard notation?

Answer 1

Indeed as mentioned in the comments and by you, this often just means almost surely: for almost every $\omega\in \Omega$ we have

$$|X_{t}(\omega)-X_{s}(\omega)|\leq C_{1}(\omega), \forall |t-s|\leq C_{2},$$

or in probability terms

$$P[|X_{t}-X_{s}|\leq C_{1}, \forall |t-s|\leq C_{2}]=1$$

$$\Leftrightarrow \int_{\Omega} 1_{|X_{t}(\omega)-X_{s}(\omega)|\leq C_{1}(\omega), \forall |t-s|\leq C_{2}} dP(\omega)=1.$$

The $C_{1}$ might just be deterministic. The $\omega$ here means that we sample a realization of the path $(X_{t})_{t\geq 0}$. See for example What is a sample path of a stochastic process for many nice answers

See for example, Kolmogorov continuity theorem, where an inequality for the moments implies an almost sure modulus of continuity with random constant.