Is the Random Element in Stochastic Processes for Compensating Unknown Actions?

AI Thread Summary
A stochastic process involves a combination of predictable actions and a random element that accounts for uncertainty. The random element serves to compensate for unknown or non-predictable actions, depending on how the underlying event space is defined. For instance, in a coin toss, the random variable can represent either the simple outcomes of heads or tails or encompass a broader context of events leading to that outcome. Understanding the dependency structure and sample paths is crucial for distinguishing between predictable and random components. This discussion clarifies the role of randomness in modeling uncertainty within stochastic processes.
FlufferNuterFSU
Messages
17
Reaction score
0
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.
 
Physics news on Phys.org
FlufferNuterFSU said:
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.

FlufferNuterFSU said:
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.
 
Back
Top