a stochastic process is a sequence of random variables whose index has the interpretation of time.
in the following, let be a polish space with borel -algebra . further, let be a probability space and let be arbitrary. we are mostly interested in the cases and an interval.
let . a family of random variables (on ) with values in is called a stochastic process with index set (or time set) and range .
(taken from Achim Klenke, 2020 chapter 9.1 processes, filtrations, stopping times)