Itô's lemma

Ito lemma 1D illustration

Itô's lemma is a fundamental result in the field of stochastic calculus, which is a branch of mathematics that deals with processes involving randomness. Itô's lemma plays a crucial role in the modeling of random processes, especially in the financial mathematics area for the modeling of stock prices, and in various other fields such as physics and engineering where systems are affected by random influences.

Overview

Itô's lemma can be considered as the stochastic calculus counterpart of the chain rule in classical calculus. However, due to the nature of stochastic processes, which include randomness, the lemma incorporates additional terms to account for the volatility in the processes. It was named after Kiyosi Itô, a Japanese mathematician who made significant contributions to probability theory and stochastic processes.

Formulation

The lemma is typically stated for a function \(f(t, X_t)\) where \(X_t\) is a Itô process. An Itô process is a type of stochastic process that can be represented as the sum of a drift term and a diffusion term, the latter being a stochastic integral with respect to a Wiener process or Brownian motion.

In its simplest form, for a one-dimensional Itô process \(X_t\) and a twice continuously differentiable function \(f(t, X_t)\), Itô's lemma states that:

\[df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}dW_t\]

where \(\mu\) is the drift coefficient, \(\sigma\) is the volatility coefficient, and \(dW_t\) represents the increment of a Wiener process or Brownian motion.

Applications

Itô's lemma is widely used in financial mathematics, particularly in the derivation of the Black-Scholes equation for option pricing. The lemma allows for the modeling of the price of financial derivatives based on the underlying asset's price dynamics, which are often modeled as stochastic processes.

In addition to finance, Itô's lemma finds applications in various areas of engineering, physics, and biology where systems exhibit randomness and can be modeled by differential equations with stochastic terms.

See Also

• Stochastic process
• Wiener process
• Black-Scholes equation
• Financial mathematics
• Kiyosi Itô