How do you get from calculus to stochastic calculus?

In summary, the path of study to understand stochastic calculus includes understanding measure-theoretic probabality theory and learning about good subsets of a given event space. Once you have that foundation, you can start to approach questions about what is the most likely outcome of an event and the probabilites of that event.
  • #1
aliendoom
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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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  • #2
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?
 
  • #3
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.?
 
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  • #4
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.
 

1. What is the difference between calculus and stochastic calculus?

Calculus is a branch of mathematics that deals with the study of continuous change and motion, while stochastic calculus is a branch of mathematics that deals with the study of random processes and how they change over time.

2. Why is stochastic calculus important in science?

Stochastic calculus is important in science because many natural phenomena, such as weather patterns and stock market fluctuations, can be modeled as random processes. By using stochastic calculus, scientists can make predictions and analyze these processes to better understand and manage them.

3. How do you apply calculus concepts to stochastic calculus?

Many of the concepts and techniques from calculus, such as derivatives and integrals, can be applied to stochastic calculus. However, in stochastic calculus, these concepts are extended to deal with random variables and processes instead of deterministic functions.

4. Is it necessary to know calculus to understand stochastic calculus?

Yes, a strong understanding of calculus is necessary to understand stochastic calculus. However, some basic knowledge of probability theory is also helpful in understanding the concepts and applications of stochastic calculus.

5. What are some real-world applications of stochastic calculus?

Stochastic calculus has a wide range of applications in fields such as finance, engineering, physics, and biology. Some examples include option pricing in finance, control systems in engineering, and modeling of diffusion processes in physics and biology.

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