Help with stochastic processes

[FlufferNuterFSU]

From my extremely small and inadequate knowledge of stochastic processes (and Wikipedia):

A stochastic process is a process in which some later state is determined by predictable actions and by a random element.

Now the question: this "random element" is this meant to compensate for unknown (non-predictable) actions or is this just a random factor for some other purpose? Hope that makes sense. Thanks.

[bpet]

A stochastic process is a process in which some later state is determined by predictable actions and by a random element.

In a way, yes. The "actions" are implicit in the underlying event space and dependency structures. In general definition of a stochastic process as a set of random variables indexed by an ordered set, the predictable part would come from our knowledge of the dependency structure and the particular sample path; the random part would be everything else we don't know about the sample path.

this "random element" is this meant to compensate for unknown (non-predictable) actions or is this just a random factor for some other purpose?

Yes - it depends on how the underlying event space is formulated. Take a coin toss as an example, where X_t=X_0 for t>0 and X_0 is a random variable that maps events to either 0 or 1. The underlying event space could be just the 2 events Heads or Tails, or it could be a description of the entire universe leading up to the point in time of the coin toss. The former model summarizes the uncertainties of the latter.

Hope this helps.