What are stochastic and deterministic processes?

During class, I occasionally mention to students that the calculus you learnt is deterministic and they go uh huh. So like what is non deterministic?! I’ll attempt to give some simple definitions and example and also touch on stochastic process.

First, a deterministic process is one that given at a particular state and an action, we only have one possible successive state to move on to. That is what we do in A-levels, since we often don’t consider any randomness in our work.

Second, a non-deterministc process is one that given at a particular state and an action, we have a set of possible successive state to move on to.

Lastly, a stochastic process is one that to begin with, has a probability of to be at a particular state, and has a finite number of states that it can transit to. It also has transition probabilities, which adds up to one.

A deterministic process is one where the present state completely determines the future state. If I make a (riskless) investment of 10,000 at 10% interest, compounded annually, then in one year's time I will have11,000, in two years’ time I will have 12,100, and so forth.  A stochastic process, on the other hand, is one where there's some randomness or uncertainty involved.  If I buy10,000 worth of \chi-company stock today, how much money will it be worth in one year’s time? In two years’ time? Who knows? This is a stochastic process.

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