A stochastic process is a process whose course depends on chance and for which probabilities for some courses are given.
A stationary stochastic process is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are present, also do not change over time.
A stationary stochastic process will exhibit the same probability distributions between:
- when taken over a large amount of samples at a certain time point, and
- when sampling a large amount along time over a single sample.
A major sin during the last financial crisis was the usage of variances in historical asset price changes to infer the risk of a large class of ABS or MBS securities. This technique is fundamental in market security analysis and has proved to be disastrous when it goes wrong. There's no rigorous proof that security prices move is a stochastic process.
Another common mistake is to see the probability of winning at a casino as the same whether it's:
- a large amount of parallel bets at the same time, or
- a single bet repeated for a large amount of times
While the same game remains random and with a fixed probability distribution function and expected value (assuming Casino doesn't change the payouts), 1,000 gamblers doing the game once is vastly different from 1 gambler doing the game 1,000 times, for among the 1,000 gamblers some will win and some will lose while for the 1 gambler, he will suffer bankruptcy at, say, the 573th trial and go down, without ever having the chance to gamble the remaining 437 times so that the numbers will work out.