Definition of Ito's Lemma

Ito's Lemma is a fundamental rule in stochastic calculus (Ito Calculus) that extends the chain rule from regular calculus to be applicable to stochastic processes like Brownian Motion. It is particularly used in evaluating functions that depend on both time and random variables, such as f(t,X(t)), which are widely used in mathematical finance.

The connection with the normal Chain Rule

In general calculus, if we have a function Y=f(t,x(t)) and x(t) is a function of t, we use the chain rule as follows:

dtdY​=∂t∂f​+∂x∂f​⋅dtdx​

But if x(t) is a stochastic process like Brownian motion or Ito process, this rule will not be sufficient because Brownian motion does not have a derivative in the usual sense.

The form of Ito's Lemma (1 variable)

If X(t) is an Ito process:

dX(t) = μ(t, X(t))dt + σ(t, X(t))dB(t)

And f(t,X(t)) is a function that has certain continuous derivatives, then Ito’s Lemma will give:

df(t,X(t))=(∂t∂f​+μ∂x∂f​+21​σ2∂x2∂2f​)dt+σ∂x∂f​dB(t)

Explanation of each term

• μ:Non-random rate of change (drift)
• σ: volatility
• 1/2σ2∂x2∂2f: An additional term not found in the normal chain rule, arising from the interaction with the variance of Brownian motion.
• Term with dB(t): random part

Example: Used with GBM

If dS=μSdt+σSdB, and you want to find d(lnS):

Using Ito's Lemma with f(S)=lnS, we get:

df=(S1​⋅μS−21​⋅S21​⋅σ2S2)dt+S1​⋅σSdB=(μ−21​σ2)dt+σdB

Which corresponds to the result used in the Geometric Brownian Motion section.

Brief summary: Ito’s Lemma is the "chain rule for stochastic equations" that is crucial for solving SDEs and leads to the derivation of the Black-Scholes formula.

Reference: Ito's Lemma

From https://www.quantstart.com/articles/Itos-Lemma/