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Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process.

My First Question

What is $\displaystyle \ \ d\int_0^t \sigma(u,T)dW(u)$? My lecture slides assert that it's equal to $\sigma(t,T)dW(t)$ but I'm not sure why. So my question is "Why"?

My Second Question

What is $\displaystyle \ \ d\int_a^t \sigma(u,T)dW(u)$, where $a \in (0,t)$.


I would appreciate careful explanation as I am not a mathematician. This is not homework, but I've put the tag on because it's on that level. Thank $\mathbf{YOU}$!

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Try to use the fundamental theorem of calculus and the fact that W(0)=0 a.s. –  Gautam Shenoy Nov 15 '12 at 16:45
    
@GautamShenoy I'm afraid that this does not assist me given my current understanding of mathematics. I don't think I can use the second fundamental theorem because i just have $d$ instead of $\frac{d}{dt}$ preceding the integral. –  Jase Nov 15 '12 at 16:50
    
It's really difficult to explain it without mathematics. To me it's like explaining color to a blind creature. No offense btw. Could you tell us your mathematical background? –  Gautam Shenoy Nov 15 '12 at 16:53
    
@GautamShenoy What I meant to ask was for you to provide some additional hints - in mathematics. I probably should've couched this in different language. I've done 1 calculus and 1 linear algebra subject at University. –  Jase Nov 15 '12 at 16:59
    
Brownian motion is studied in a graduate level course on stochastic processes. So I'm wondering where did you encounter it? In a research problem perhaps? –  Gautam Shenoy Nov 15 '12 at 17:01

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