A fundamental tool of stochastic calculus , known as Ito's Lemma

A fundamental tool of stochastic calculus, known as Ito's Lemma, allows us to derive the Black-Scholes equation in an alternative manner.

an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process

In mathematics, Itô's lemma is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process.

the mathematical tool

Ito's Lemma is the mathematical tool we can use to prove that a potential solution to our SDE really satisfies the SDE.

a stochastic analogue of the chain rule of ordinary calculus

Ito's Lemma is a stochastic analogue of the chain rule of ordinary calculus.

the version of the chain rule or change of variables formula which applies to the Itô integral

Itô's lemma is the version of the chain rule or change of variables formula which applies to the Itô integral.

the mathematical tool we can use to prove that a potential solution to our SDE really satisfies the SDE

Ito's Lemma is the mathematical tool we can use to prove that a potential solution to our SDE really satisfies the SDE.

the stochastic version of the chain rule of standard calculus

To answer the more general question that seems to be giving you trouble, Ito's lemma is the stochastic version of the chain rule of standard calculus.

a critical result in the field of stochastic calculus

Ito's Lemma is a critical result in the field of stochastic calculus, and stochastic calculus allows us to differentiate functions of stochastic processes, which are processes with inherent randomness.

a cornerstone of quantitative finance and quantitative finance

Ito's Lemma is a cornerstone of quantitative finance and quantitative finance is intrinsic to the derivation of the Black-Scholes equation for contingent claims (options) pricing.

otherwise known as the Ito formula

Ito’s lemma, otherwise known as the Ito formula, expresses functions of stochastic processes in terms of stochastic integrals.