Mathematical philosophical questions about the general theory of stochastic processes. After 6 months spent on what is termed the "general theory" of stochastic processes and after having worked out many nuances of the field, I realized
that:


*

*The general theory is beautiful mathematics;

*Many times one might think he or she understood why a certain notion is being developed, only to have to go back to square one and re-start thinking;

*Some results are extremely difficult if one plays honestly and learns more than the statements about  the Projection and the Measurable Selection Theorems. I mean, here you are, all proud of having more or less mastered measure theory only to get pushed into analytic sets theory or at least capacity theory. It gives a real meaning to Aristotle's "The more you know, the more you know you don't know".


But here is my question to everyone who would like to pitch in. It is clear that, despite its name, the general theory is quite restrictive actually being the study of stochastic processes indexed by ${\bf R}^+ .$ But, if you
had to motivate a student to invest in learning it, how would you motivate him or her? I know that it is necessary if one wants to study the most general form of stochastic calculus, but what else? I personally have a strong dislike of stochastic processes when it all comes down to computations and love being able to follow their sample paths a la Levy, so to speak. But this sense of beauty cannot be all there is to it.
So here is the question: if you had to write 10-15 (even 20 if need be) lines to introduce the general theory of stochastic processes  and make it
as enticing as possible, what would you write? I surely have had more trouble communicating what I was learning than the learning itself
Thank you in advance for any input.
Maurice 
 A: It would be easy to say that any abstract set $A$ and any set $\{X_t, t\in A\}$ of random variables together form a stochastic process. However, we ought to prove then that for any sequence of $t_1,t_2,\cdots, t_n \in A$ and $t'_1,t'_2,\cdots, t'_m \in A$ $X_{t_1},X_{t_2}, \cdots X_{t_n}$ and $X_{t'_1},X_{t'_2}, \cdots X_{t'_m}$ are random vectors on the same probability space.
The question is as follows: what mathematical structure we have to assume regarding the set $A$ and regarding the set of distribution  functions $\mathscr F=\{F\}$ for which it is true that there exists a probability space $$[\Omega,\mathscr A, \mathbb P]$$ such that for any sequence of $t_1,t_2,\cdots, t_n \in A$ there exist distribution functions  $$F_{X_{t_1},X_{t_1},\cdots,X_{t_n} }(x_1,x_2,\cdots,x_n)=\mathbb P(X_{t_1}<x_1,X_{t_1}<x_2,\cdots,X_{t_n}<x_n)$$ 
in $\mathscr F$ or, rather, measurable sets
$$\{\omega \in \Omega:X_{t_1}(\omega)<x_1,X_{t_1}(\omega)<x_2,\cdots,X_{t_n}(\omega)<x_n \}$$
in $\mathscr A$ in such a way that no contradiction could arise.
As an example, consider the set of real functions $\{x=mx, m\in[0,1]\}$ and let $\mu$ be a random variable uniformly distributed on the interval $[0,1]$. Is it obvious that that there exists a probability space $[\Omega,\mathscr A, \mathbb P]$ such that $X_t(\omega)=\mu(\omega)t$ is a stochastic process?
A: Thank you guys. I appreciate your enthusiasm and I surely can say that I never thought of stochastic processes as an approach to picking up girls spicing the conversation with the right measure theory results. 
But, my question was specific about the general theory, that branch introduced by Paul-Andre Meyer and the Strasbourg School he ran for many years. If I try to describe what it is about I end up with one pager of concepts and notions that mean nothing to anyone who does not know it already. I can find interesting way to motivate a student to approach almost any subject in probability at large. But the general theory, aside from the application to stochastic calculus, I really seem to find no way to get anything good accomplished. 
A: Will you end up making a profit…or loosing it all?  
Random process theory deals with likelihoods, expectations, and convergence phenomena of experiments that produce an infinite collection of random outcomes.  This includes sequences such as $\{X_1, X_2, X_3, \ldots\}$ that represent the random profits of a repeated investment strategy.  A random process is also called a stochastic process.  There are wide applications of the general theory of these processes. Economists use random processes to treat unknown prices and demands.  Communication engineers use random processes to design communication systems that are robust in the presence of disturbances.  Physicists use it to study interacting particles, and casinos use it to design games of chance. 
You, too, should study the theory of random processes, so that you can be as cool as economists, physicists, engineers, and gamblers!  
