 Sample path

For a stochastic process $X_t$ and a fixed sample space outcome $\omega$, its sample path is the function $t\mapsto X_t(\omega )$. One row of observed fair bits, one trajectory of a random walk, or one staircase of Poisson process arrival counts is a sample path. Fixing $t$ instead gives a random variable ranging across possible histories. The same two-variable object $X_t(\omega )$ therefore describes both observations across outcomes and evolution within an outcome.

Statements about continuous, càdlàg or Hölder continuous function sample paths concern these functions simultaneously in time. They differ from assertions made at each fixed time separately. For example, take $U$ with a uniform distribution on $[0,1]$ and set $X_t=1_{\{t=U\}}$. For every fixed $t$, $X_t=0$ almost surely, but every sample path has a single spike. Thus this process is a modification of a stochastic process that is identically zero, while the two are not indistinguishable stochastic processes. Continuity of both sets of sample paths rules out this discrepancy: equality at a countable dense set then implies equality everywhere. This is why path regularity is an additional part of constructing a continuous-time model.