Ito Formula

Contents (hide)

1. Overview
2. The formula
3. Examples
1. 3.1 Solution of Ito integrals
2. 3.2 Asset prices

1. Overview

Ito processes are closed under functional transformations. That is to say if $\fs4 X_t$ is an Ito process then a (twice differentiable) $\fs4 g(t,x)$ will define a new Ito process $\fs4 Y_t = g(t,X_t)$ . The celebrated Ito formula will give the dynamics of $\fs4 Y_t$ given the stochastic differential equation that is satisfied by $\fs4 X_t$ .

The Ito formula plays the role of the chain rule for stochastic calculus. Just like the chain rule it can be used to simplify SDEs and derive their solutions. It has been extensively used in mathematical finance to produce the celebrated Black-Scholes partial differential equation.

2. The formula

We consider an Ito process

$\fs4 d X_t = \mu(t, \omega) d t + \sigma(t, \omega) d B_t$

A (twice differentiable) function $\fs4 g(t,x)$ will define a new Ito process $\fs4 Y_t = g(t, X_t)$

$\fs4 d Y_t = \frac{\partial}{\partial t}g(t, X_t) d t + \frac{\partial}{\partial x} g(t, X_t) d X_t + \frac{1}{2} \frac{\partial^2}{\partial x^2}g(t, X_t) (d X_t)^2$

where $\fs4 (d X_t)^2$ is computed using

$\fs4 d t \cdot d t = d t \cdot d B_t = 0 \text{, and } (d B_t)^2 = d t$

3. Examples

3.1 Solution of Ito integrals

Consider a Brownian motion $\fs4 B_t$ , and the function $\fs4 g(t, x) = \frac{x^2}{2}$ ; the process for $\fs4 Y_t = \frac{B^2_t}{2}$ will satisfy

$\fs4 d Y_t = 0 \cdot d t + B_t d B_t + \frac{1}{2}\cdot 1\cdot (d B_t)^2 = B_t d B_t + \frac{d t}{2}$

In other words

$\fs4 d \left( \frac{1}{2} B^2_t \right) = \frac{1}{2} d t + B_t d B_t$

This means that the stochastic integral

$\fs4 \int_0^t B_s d B_s = \frac{1}{2}B_t^2-\frac{1}{2}t$

3.2 Asset prices

Say that an asset price, $\fs4 S_t$ , satisfies the SDE for the Geometric Brownian motion

$\fs4 d S_t = \mu S_t d t + \sigma S_t d B_t$

From the natural logarithm $\fs4 g(t, x) = \log x$ we retrieve the process for $\fs4 s_t = \log S_t$

$\fs4 d s_t = \left( \mu - \frac{1}{2}\sigma^2 \right) d t + \sigma d B_t$

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