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Quadratic Variation in Itô’s Formula for Geometric Brownian Motion

Article Quant Q&A · Author: zuiqo
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Summary

The document asks about the bracket notation in an Itô expansion for the logarithm of a geometric Brownian motion. In the expression for the change in the log process, the term involving the process in square brackets denotes its quadratic variation. This captures the accumulated squared increments of a stochastic process, which contributes to Itô’s formula for functions of processes with random fluctuations.

The answer identifies the notation but gives no further derivation, alternative conventions, or detailed explanation of how quadratic variation yields the familiar drift adjustment for the log of geometric Brownian motion. The example is therefore a brief notation clarification rather than a full treatment of stochastic calculus. It is useful background when reading continuous-time asset models and their transformations.

Key ideas

  • Square brackets around a process in this context denote its quadratic variation.
  • Quadratic variation accounts for accumulated squared process increments in stochastic calculus.
  • Itô’s formula includes a quadratic variation term when transforming stochastic processes.
  • For log geometric Brownian motion, this term contributes to the transformed process dynamics.

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# Bracket-Notation in SDEs


# 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!

## Answer by vonjd (score 4, accepted)

https://quant.stackexchange.com/a/8377

It is a notation for quadratic variation of a stochastic process.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.