How do you get from calculus to stochastic calculus?

[aliendoom]

What is the path of study to understand stochastic calculus? I bought the book "Elementary Stochastic Calculus with Finance in View" (Mikosch) because it was touted as a non rigorous introduction to stochastic calculus, and I spent three days trying to decipher the first two pages. :(

 

[Lonewolf]

What level of mathematics education have you reached? You probably ought to have at least some level of familiarity with measure-theoretic probabality theory. What was the content of the first two pages?

 

[aliendoom]

What level of mathematics education have you reached?

I'm currently brushing up on forgotten calculus and statistics, so I'm afraid I haven't retained very much of my math education.

You probably ought to have at least some level of familiarity with measure-theoretic probabality theory.

How do I get from basic calculus and statistics to measure-theoretic probability? Do you have any book recommendations. I guess I would be looking for the least rigorous books available. A list of courses might be helpful as well, so I could see the progression and how far away I am.

What was the content of the first two pages?

The book starts out its description in the setting of a coin flip experiment. It says, if we let "heads" equal 1 and "tails" equal 0, then we get a random variable:

$X=X(\omega)\epsilon\{0,1\}$

where $\displaystyle\omega$ belongs to the outcome space $\Omega=\{heads, tails\}$

After I deciphered the notation, that seemed straightforward enough. But, then under the innocuous subheading:

"Which are the most likely $X(\omega)$, what are they concentrated around, what are their spread?

the book says that to approach those problems, one first collects "good" subsets of $\Omega$ in a class F, where F is a $\sigma$-field. Such a class is supposed to contain all interesting events. Certainly, {w:X(w)=0}={tail} and {w:X(w)=1}={head} must belong to F, but also the union, difference, and intersection of any events in F and its complement the empty set. If A is an element of F, so is it's complement, and if A,B are elements of F, so are A intersection B, A union B, A union B complement, B union A complement, A intersection B complement, B intersection A complement, etc.

Whaaa? What's all that $\sigma$-field stuff got to do with the probabilites of X(w)? Also, if A and B are a member of a class F, isn't A union B also automatically a member of the class F, as well as A intersection B, etc.?

 

[Lonewolf]

Take a peek at http://www.math.uconn.edu/~bass/lecture.html. I had a look around amazon and couldn't find anything like a non-rigourous book on the subject, and most of the books seem a bit pricey considering that you'll only be reading one or two chapters from them. Have a look at those notes and see how you get on.