Ito's Lemma gives a formula for the differential of a function applied to a stochastic process driven by Brownian motion, extending the ordinary chain rule of calculus with an extra term arising from the process's quadratic variation, a feature absent from ordinary smooth functions. Named for Kiyoshi Ito, it is the foundational computational tool of stochastic calculus, underlying applications from physics to mathematical finance.
Facts
StatementFor an Ito drift diffusion process dXt = mu_t dt + sigma_t dBt and a twice differentiable function f(t,x), the differential of f(t,Xt) is (df/dt + mu_t df/dx + sigma_t squared over 2 times d2f/dx2) dt + sigma_t df/dx dBt. 1 Classification
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Source Itô's lemma (Wikipedia)
Proved By
Source Itô's lemma (Wikipedia)
Sources
1. Ito's lemma, Wikipedia
Introduction, description of the result
the stochastic calculus counterpart of the chain rule
Lead section
This result was discovered by Japanese mathematician Kiyoshi Itô in 1951.
View the SourceItô's lemma (Wikipedia)
Proved By: Kiyosi Itô, Lead
This result was discovered by Japanese mathematician Kiyoshi Itô in 1951.
In Branch: Probability and Statistics, Lead paragraph
In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process.
In Group: Stochastic Processes and Martingales, Lead section
to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart
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