Itô's Lemma
From the book
Chapter 19: Stochastic Differential Equations. In the chapter mind map this icon labels Itô's Lemma:Stochastic Chain Rule. The discussion below is excerpted and lightly edited from § Itô's Lemma in Mathematics for AI and Machine Learning.
Consider the geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t$ and the function $f(x) = \log x$.
Substituting $dX_t = \mu X_t dt + \sigma X_t dW_t$ and $(dX_t)^2 = \sigma^2 X_t^2 dt$:
Key observation: The drift in $\log X_t$ is $\mu - \frac{\sigma^2}{2}$, not $\mu$. This is the Itô correction—the noise reduces the effective drift due to the convexity of the logarithm function.
What this drawing shows
What you see. Itô chain-rule box fixed; the red Itô correction term $d[x]_t = g^2\,dt$ fades in to show the second-order stochastic correction.
In the mind map. Chapter 19 — Itô's Lemma:Stochastic Chain Rule. See From the book above for definitions, figures, and worked examples.
Where to read next
Read the full definitions, figures, and worked examples in Chapter 19: Stochastic Differential Equations — see the mind-map node Itô's Lemma:Stochastic Chain Rule.
