Definition:Modification of Stochastic Process

Definition

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\struct {E, \mathcal E}$ be a measurable space.

Let $\sequence {X_i}_{i \in I}$ and $\sequence {\widetilde X_i}_{i \in I}$ be $E$-valued stochastic processes.

We say that $\sequence {\widetilde X_i}_{i \in I}$ is a modification of $\sequence {X_i}_{i \in I}$ if and only if:

$\widetilde X_i = X_i$ almost surely

for each $i \in I$.

Sources

• 2016: Jean-François Le Gall: Brownian Motion, Martingales, and Stochastic Calculus ... (previous) ... (next): Definition $2.6$