Definition
A process is a function that takes on possibly different values at different times. That is, it is a function
{% X(t) : t \rightarrow X_t %}
{% t %} may either take on values from a discrete set such as {% {0,1,2,3,4,5...} %}
or from an interval {% [a,b] %}.
A stochastic (or random process) is one where there are a collection of functions, indexed by {% \omega \in \Omega %}
{% X(\omega) = t \rightarrow X_t(\omega) %}
Notions of Equality
There are several notions of equality between two processes.
- Modification:
{% Y %} is a modification of {% X %} if {% \mathbb{P}[Y_t = X_t] = 1 %} for all {% t %} - Indistinguishable: - two processes are indistinguishable if the all paths are equal, except for a set of paths with measure zero.