Itô’s Formula and the Quadratic Variation of Geometric Brownian Motion

Summary

This document raises questions about applying Itô calculus to geometric Brownian motion, including why the squared Brownian increment contributes a variance term and whether it is reasonable to remove the random component from the stochastic differential equation. It also asks how to apply Itô’s formula when the process itself appears in the coefficients.

The text contains the questions but no answer or worked derivation. It does not establish the claimed differential relationship or explain the role of quadratic variation, so readers should treat it as a prompt for learning rather than a complete lesson. Its useful subject is the distinction between stochastic increments and ordinary differentials when analyzing GBM.

Key ideas

• Geometric Brownian motion includes both drift and a random Brownian component.
• The document asks how Itô’s formula applies when a function depends on the stochastic process.
• It raises quadratic variation as the reason squared Brownian increments contribute a time term.
• No derivation or answer is supplied, so the questions remain unresolved.

Tags

# Geometric Brownian Motion SDE


# Geometric Brownian Motion SDE












I recently saw the clip : GBM which quantpie made.

here is the link https://www.youtube.com/watch?v=98xF6b0PZpo

In the time 1:33 of the clip, it naturally said $dX^2 = σ^2X_t^2dt$

To proof this it assumes that there are no random terms. So it assumes $dX_t = μX_tdt$

My thinking is : $dX_t = μX_tdt+σX_tdB_t$ $dX_t = f_tdt + f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt = df$ (when it is assumed that $f = X_t$)

I mean, I tried to solve the equation using the ito's integral formula but I don't know what to do, because of the term : $X_t$. Should I use ito's formula for ito process..?

To organize my question

- Why $dX^2 = σ^2X_t^2dt$?

- Why the assumption $dX_t = μX_tdt$ is reasonable? there was actually random term. And how it could be verified that there are no random term?

- How can solve differential equation with the formula $df = f_tdt + f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$

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