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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.2012-11-15
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    @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.2012-11-15
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    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?2012-11-15
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    @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.2012-11-15
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    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?2012-11-15
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    @GautamShenoy Ok well I guess I've done 2 calculus/analysis subjects and 1 linear algebra, forgot the first 1 I did. It's in a finance undergraduate degree, they decided to chuck in a quantitative finance course which is all stochastic calculus with a lil measure theory and probability theory.2012-11-15
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    @did Can you supply the correct answer please?2012-12-13

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