Recently I have re-take the course Stochastic Calculus because it must be very useful and I want to compensate for my failure 3 years ago since I didn't acquire the knowledge of this course at that time.
After spending some time reading the materials, gradually I remembered what I learned three years ago. Those concepts and theorems come to my mind just like old memory. There are two main difficulties in studying stochastic calculus, the concepts are very abstract and the symbols people use are misleading, which worsen the first problem. I was tortured by these two difficulties last time. Now since this is the second time to study the course,I got new understand of those concepts and theorems and some old confusion which puzzled me before now can be clearly explained. Today I want to describe my understanding of the most basic concept in Stochastic Calculus-Stochastic Process.
What is random variable? Based on my previous experience,series such as {1,3,6,323,6,885,33,56,2,7……….} is a random variable and that’s all. I can’t imagine any random variable beyond this kind of series of number.
OK, now look at what is the definition of random variable int he textbook…
Oh damn it! What the hell is this? What’s the relation ship between those obscure words and the friendly {1,3,6,323,6,885,33,56,2,7……….}?
OK, come down. After overcoming the initial pulse of nerves,I begin to figure the way out because I know that mathematicians do not waste their time to write nonsense. It is said that X is a family of Xt, which is a mapping from Omega to E, en, X seems to be a mapping from (Omega, t) to E. In a simple example, let E be a real value space R, I can imagine a stochastic process be a two dimensional space, which is something like:
This is the picture of stochastic process I acquired last time I study this course. Indeed it can explain something and I learned a lot from it. However, there is something that haunts me when I think about this picture. I know it is not precise but I don’t know why.
This time when I look at the picture, suddenly I realized the problem is that both the two axis can only present real valued space R, or time T, none of the axis can present a more general space Omega. Omega is the space of Events we care about and the operations within the space is only union and intersection.