Can anyone please give a simple example of what is a stochastic process?
I got confused between random variables and stochastic process. I've read somewhere that stochastic process is a collection of random variables. A real life example might help to understand what that means.
Thanks a lot.
The following may help or not, it is a particular simplified example, that tries to show why "a collection of random variables may be interesting". (From a modelling point of view.)
The "time interval" $T$ can be taken to be one of the following while dealing with stochastic processes:
$T$ is the finite set consisting of $0,1,2,\dots ,N$, where $N$ is some fixed natural number.
$T$ is $\Bbb N$ (or $\Bbb Z$).
$T$ is $\Bbb R_{\ge 0}$ (or $\Bbb R$).
Of course, we take here the first case, i am working with $N=3$ which is "complicated enough", so $T=\{0,1,2,3\}$. Our probability space is $\Omega=\Omega_0^{\{1,2,3\}}$, where $\Omega_0=\{H,T\}$ models the outcome of a coin toss, so $\Omega$ has eight elements, $HHH$, $HHT$, $HTH$, and so on till $TTT$.
Now consider a game, the win in the one or the other case is given by the following randomly generated set of numbers:
(This is only may way of getting some numbers quickly typed. Consider only the above numbers.)
This models a "random variable" $X_3$. It is "known at time point $3$", i.e. when we know all components $\omega_1,\omega_2,\omega_3$ (and thus also "all $\omega$") of some $\omega\in\Omega$.
But at point $2$, knowing only $\omega_1,\omega_2$, we have some "uncertain situation", e.g. knowing only $HH$ we may expect $HHH$ or $HHT$ in the final, winning $20$ or $32$, so the mean is $26$. Doing the same with all the other cases, we get a random variable with values
These are the "fair prices" for playing till the end, from a given situation. They describe a "random variable" $X_2$, which is "known in $2$" (in the model, mathematically we describe a sigma-algebra so that $X_2$ is measurable w.r.t. it). We have for instance $X_2(THT)=X_2(THH)=X_2(TH*)=26$.
We can go one increment back in time, then the "fair prices" are updated because of the less information:
We get a random variable $X_1$ with the above values.
So the initial price is $X_0$, the constant
33.5.Now back to the question. Yes, generally speaking, a stochastic process is a collection of random variables, indexed by some "time interval" $T$. (Which is discrete or continuous, usually it has a start, in most cases $t_0:\min T=0$.)
But it also has an ordering, and the random variables in the collection are usually taken to "respect the ordering" in some sense. In fuzzy words, if $s\le t$ on the time line, we expect that $X_s$ can be obtained from $X_t$ by "restricting the information". (We use conditional expectations. The "information in $t$" is declared by some $\sigma$-algebra $\mathcal F_t$ from the start. We also have an included (less information) $\sigma$-algebra $\mathcal F_s$ for the time point $s$. Then $X_s$ is obtained from $X_t$ by taking conditional expectation of $X_t$ w.r.t. $X_s$.)
In our example, we have a $\max T=3$ (in our case), so the whole information of the process can be extracted from the correspondin random variable ($X_3$ in our case), but if $T=\Bbb N$ or $T=\Bbb R$, the "finish" is not known. We simply go with the time and need all pieces of the process.
Things become more and more complicated, because the theory puts some special (families of) distributions of probability on the individual variables, and because we may "stop" the process not only at some "fixed point", e.g. $2$, getting $X_2$, but also at some random point. (For instance when the first two $H$ tosses were seen. This is a "stopping time", an "optional time" $\tau$, they write $X_\tau$ for the corresponding random variable.)
So the definition of a "general random process" is indeed general, take a collection of random variables indexed by some time points, but the theory goes into details in "interesting cases" modeling the evolution of information with the time, and plays "special games" with random times.