# Bracket-Notation in SDEs

I often come across the following notation in my script, and I have not found it anywhere else. While our lecturer insists it is of utmost importance to write this way in his exams, he yet failed to explain why...

## Setup

From a geometric Brownian motion $$dX_t = \mu X_tdt+\sigma X_t dW_t$$ apply Ito to $$f(X_t,t) = ln X_t =: Y_t$$ and get

$$dY_t = \frac{1}{X_t}dX_t - \frac{1}{2X_t^2}d[X]_t$$

## Actual Question

Why is X in squared brackets in the second term of the RHS?

What is the specific background to write this way, and how is it different from other notations?

All texts I've worked with so far (pure finance, except Oksendal) were able to work without this. What am I missing here?

I appreciate your help!

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Well I assume it's his personal way of writing squared Brownian motions, however he never defines it. Does this notation by any convention carry additional information? – phi Jul 2 '13 at 18:05
This is a standard notation for the quadratic variation of a stochastic process: en.wikipedia.org/wiki/Quadratic_variation – olaker Jul 2 '13 at 20:36
For a local martingale there is a difference (in general) between $\langle M\rangle$ and $[M]$. The two processes coincide, if $M$ is continuous. – user8 Jul 3 '13 at 8:14
Thank you olaker. I edited my answer accordingly for future reference. – vonjd Jul 4 '13 at 9:48
Thanks to all of you! – phi Jul 4 '13 at 12:56