A discrete stochastic process is simply a sequence of random variables. For the process to be interesting, these variables should depend on each other in interesting ways: for example, could be a stochastic process where is the position of a random walk after steps. A continuous stochastic process is a set of random variables indexed by real numbers instead of integers: the most famous example is Brownian motion. This article contains links to articles about proofs of facts about stochastic processes and proofs that use stochastic processes.
A knowledge of the basics of probability theory.