What is a Stochastic Integral and How Does it Differ from a Regular Integral?

[courtrigrad]

Hello all

Let's say we define a stochastic integral as:
$$W(t) = \int^{t}_{0} f(\varsigma)dX(\varsigma) = \lim_{n\rightarrow\infty} \sum^{n}_{j=1} f(t_{j-1})(X(t{j})) - X(t_{j-1}))$$ with $$t_{j} = \frac{jt}{n}$$ IS this basically the same definition as a regular integral?

Also if we have $$W(t) = \int^{t}_{0} f(\varsigma) dX(\varsigma)$$ then does $$dW = f(\varsigma) dX$$?

Thanks

 

[dextercioby]

In the first integral i can see a strong resemblence with the Riemann sum...As for the second (and for the first too),who's zeta...?

Daniel.

 

[polyb]

The Weiner process the one you are looking for and luckily old Norbert worked it out for us. This really becomes more statistical than anything because we have to talk about the average or standard deviation of each step in the integral. It has been a little while and I don't have any notes with me at the present moment but Norbert is the man to look into to wrap your mind around stochastic integrations!

 

[courtrigrad]

zeta is a variable corresponding to time