Itô’s Lemma for Functions of Stochastic Processes

Summary

The document explains Itô’s lemma as the stochastic counterpart of the ordinary chain rule. It starts from a drift-diffusion process driven by Brownian motion and describes how to find the differential of a sufficiently smooth function that depends on both the process and time. The resulting expression includes partial derivatives with respect to time and the process, plus a second-order term arising from the process’s randomness.

The explanation gives the differential rules used in the derivation: squared time increments and mixed time–Brownian increments vanish, while the squared Brownian increment contributes a time increment. It places the lemma in quantitative finance, noting its role in deriving the Black–Scholes equation. The document provides the theorem statement but no worked derivation or numerical example, and it assumes familiarity with Brownian motion and stochastic differential equations. It presents the standard framework rather than discussing extensions or the limits of the model assumptions.

Key ideas

• Itô’s lemma extends the chain rule to functions of stochastic processes and time.
• The differential includes a second-order term because Brownian increments have nonzero quadratic variation.
• Time increments squared and mixed time–Brownian increments are treated as zero in the stated rules.
• The lemma is a component of the derivation of the Black–Scholes equation.
• The exposition assumes prior knowledge of Brownian motion and stochastic differential equations.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.