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Here I have this question

(i) state Ito's formula

(ii) hence or otherwise show that

$\int^t_0B_s dB_s = \dfrac{1}{2}B^2_t -\dfrac{1}{2} t$

(iii) define the quadratic variation $Q(t)$ of Brownian motion over [0,t], given that $Q(t) = t$, use this result to prove (ii)

I can do everything up to the last bit of (iii), how can quadratic variation tell you this relationship?

disclaimer: this is not homework. I am trying to help a friend preparing for an exam and this was a past paper question.

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Please change the title of this "question" (currently being "Exercise on stochastic calculus" to s.th. meaningful (e.g. use words like "Ito formula" and "Quatratic Variation"). –  Christian Fries May 11 '13 at 18:37
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2 Answers

up vote 3 down vote accepted

"Like" Ito: $$d (B^2) = B dB + B dB + dB dB$$

That is $$B dB = \frac{1}{2} d (B^2) - \frac{1}{2} dB dB$$

Integrate. Last term is 1/2 the quadratic variation.

I understand the questions as follows: In iii) one has to define what $dB dB$ stands for and one has to "proof" the first line in my answer. In ii) one may use Ito to "know" that $dB dB = dt$.

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+1 sweet use of Ito product rule. –  Veeken May 11 '13 at 19:33
    
how is this any different to the answer for part (ii) where you are asked to show that was the case using ito's formula? –  Lost1 May 11 '13 at 23:45
    
Ito actually tells you that $d(B^2) = BdB + BdB + dt$. So ii) might mean: Proof it using Ito, while iii) means proof it in an elemantary by repeating the proof of Ito for the special case of $B^2$. –  Christian Fries May 13 '13 at 13:53
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i picked this off from Shreve.

Start with the definition of sampled quadratic variation: (1) $\frac{1}{2}Q_\pi = \frac{1}{2}\sum\nolimits_{j=0}^{n-1} (W_{j+1}) - W_j)) ^2$ where $\pi$ = {0,1,2...,n} is a partition of $[0,T]$ (Note we took $\frac{1}{2}$ of both sides for reasons that will be clear in the next line.) Now we know (1) is equal to $\frac{T}{2}$, but we also know by simple algebra that
(1) =$\frac{1}{2}W_n^2 + \sum\nolimits_{j=0}^{n-1} W_j(W_j - W_{j+1})$.

All that remains to show the result is to make it rigorous in the sense that we're approximating a Brownian motion with a discretized version that converges in the limit as $n \to \infty$; we're also approximating the ito integral with sums, which also converge in the limit. Will leave this to you to iron out a bit further. Again, reference Shreve's notes if you don't have his excellent texts (google search :steve shreve notes)

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What i am asking is, how do you know the expression for part (ii) using knowledge of quadratic variation? –  Lost1 May 11 '13 at 23:46
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