Itō's lemma
In mathematics, Itô's lemma is an identity used in Itô calculus to find the differential of a time-dependent function of a . It serves as the stochastic calculus counterpart of the chain rule. It can be heuristically derived by forming the Taylor series expansion of the function up to its second derivatives and retaining terms up to first order in the time increment and second order in the Wiener process increment. The lemma is widely employed in mathematical finance, and its best known application is in the derivation of the Black–Scholes equation for option values.
Itô's lemma, which is named after Kiyosi Itô, is occasionally referred to as the Itô–Doeblin theorem in recognition of posthumously discovered work of Wolfgang Doeblin.
Note that while Itô's lemma was proved by Kiyosi Itô, Itô's theorem, a result in group theory, is due to Noboru Itô.
A formal proof of the lemma relies on taking the limit of a sequence of random variables. This approach is not presented here since it involves a number of technical details. Instead, we give a sketch of how one can derive Itô's lemma by expanding a Taylor series and applying the rules of stochastic calculus.
Assume Xt is a Itô drift-diffusion process that satisfies the
where Bt is a Wiener process. If f(t,x) is a twice-differentiable scalar function, its expansion in a Taylor series is
Substituting Xt for x and therefore μt dt + σt dBt for dx gives
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