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How do you get from calculus to stochastic calculus?

by aliendoom
Tags: calculus, stochastic
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aliendoom
#1
Oct17-05, 01:54 AM
P: 29
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. :(
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Lonewolf
#2
Oct17-05, 05:27 PM
P: 333
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
#3
Oct19-05, 04:04 AM
P: 29
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:

[itex]X=X(\omega)\epsilon\{0,1\}[/itex]

where [itex]\displaystyle\omega[/itex] belongs to the outcome space [itex]\Omega=\{heads, tails\}[/itex]

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

"Which are the most likely [itex]X(\omega)[/itex], what are they concentrated around, what are their spread?

the book says that to approach those problems, one first collects "good" subsets of [itex]\Omega[/itex] in a class F, where F is a [itex]\sigma[/itex]-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 [itex]\sigma[/itex]-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
#4
Oct19-05, 08:46 AM
P: 333
How do you get from calculus to stochastic calculus?

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.


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