Path continuity of stochastic processes

\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

“When are the paths of a stochastic process continuous?” is a question one might like to ask. But things are never so simple in stochastic process theory. Continuity is not unambiguous here, and probability theory is not so gracious as to be intuitive about this; we need to ask more precise questions. If we are concerned about whether the paths sampled from the process are almost-surely continuous functions then we probably mean something like:

“Does the process \(\{\rv{x}(t)\}_t\) admit a modification such that \(t\mapsto \rv{x}(t)\) is in Hölder-continuous with probability 1?”

More generally there are many notions of continuity of stochastic processes.

Kolmogorov continuity theorem.

The Kolmogorov continuity theorem gives us sufficient conditions for Hölder-path-continuity of the process based on how rapidly moments of the process increments grow. Question: What gives us sufficient conditions? Lowther is good on this.

Stochastic continuity

An entirely different thing. Feller-continuity etc.

Continuity entailments