Definition:Modification of Stochastic Process
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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 $2.6$