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 càdlàg modification of a stochastic process is a modification of a stochastic process whose paths are càdlàg on one event of probability one. Equality almost surely at every separately fixed time does not imply indistinguishability of stochastic processes when the original process lacks path regularity. A stochastic process starting at zero, with independent increments, stationary increments, and stochastic continuity, has a càdlàg modification and is a Lévy process under the intrinsic definition.

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.