What is a right continuous(or left continuous ) stochastic process?

Question

I understand the concept of left and right continuity in a real line , but how is it defined for a stochastic process? Do we fix $\omega$ and check the continuity of the path as time evolves, or is it something else?

Answer 1

Yes, indeed. A stochastic process $(t,\omega) \mapsto X(t,\omega) = X_t(\omega)$ is called right-continuous (left-continuous) if, and only if, $$t \mapsto X_t(\omega)$$ is right-continuous (left-continuous) for all $\omega \in \Omega$.

Answer 2

A stochastic process $(\mathrm{X}_t)_{{t} \in \mathbb{R}+}$ is right-continuous if for all $\omega \in\Omega, $ there is a positive $\varepsilon$ such that $X_s(\omega)=X_t(\omega) $ holds for all s, t satisfying $t \leq s \leq t + \varepsilon.$ This is based on these lecture notes. This definition applies to all stochastic processes that are indexed over the (non-negative) real numbers; however, other notions of continuity are listed on Wikipedia.

This leaves open the question on how this notion of continuity relates to the notion of right-continuity for functions $ f:\mathbb R\to\mathbb R$ from real numbers to real numbers.