Modification of a stochastic process

Processes $X,Y$ on the same probability space and index set are modifications of each other if $\mathbb{P}(X_t=Y_t)=1$ for each fixed index $t$. This differs from indistinguishability of stochastic processes, which requires equality at all indices on a single event of probability one. Equality at a countable dense set upgrades to indistinguishability when both processes have continuous paths.

A continuous modification of a stochastic process is a modification of a stochastic process whose paths are continuous outside a single null event. For a process originally given only on a countable dense set, the analogous construction gives a continuous extension agreeing simultaneously at all original indices. The Kolmogorov continuity theorem and dyadic increment chaining are standard ways to obtain it.