Ito's Formula
Theorem 1
Given an Itô process $\left\{ X_{t} \right\}_{t \ge 0}$, $$ d X_{t} = u dt + v d W_{t} $$ for a function $V \left( t, X_{t} \right) = V \in C^{2} \left( [0,\infty) \times \mathbb{R} \right)$, let $Y_{t} := V \left( t, X_{t} \right)$, then $\left\{ Y_{t} \right\}$ is also an Itô process and the following holds: $$ \begin{align*} d Y_{t} =& V_{t} dt + V_{x} d X_{t} + {{ 1 } \over { 2 }} V_{xx} \left( d X_{t} \right)^{2} \\ =& \left( V_{t} + V_{x} u + {{ 1 } \over { 2 }} V_{xx} v^{2} \right) dt + V_{x} v d W_{t} \end{align*} $$
- $C^{2}$ is a class of functions that are twice differentiable and their derivatives are continuous.
- $\displaystyle V_{t} = {{ \partial V } \over { \partial t }}$, $\displaystyle V_{x} = {{ \partial V } \over { \partial X_{t} }}$, and $\displaystyle V_{xx} = {{ \partial^{2} V } \over { \partial X_{t}^{2} }}$.
- $\left( d X_{t} \right)^{2} = d X_{t} \cdot d X_{t}$ is calculated according to the following Itô multiplication table. $$ \begin{align*} \left( dt \right)^{2} =& 0 \\ dt d W_{t} =& 0 \\ d W_{t} dt =& 0 \\ \left( d W_{t} \right)^{2} =& dt \end{align*} $$ Therefore, $$ \begin{align*} \left( d X_{t} \right)^{2} =& \left( u dt + v d W_{t} \right) \left( u dt + v d W_{t} \right) \\ =& u^{2} \left( dt \right)^{2} + 2 uv dt d W_{t} + v^{2} \left( d W_{t} \right)^ {2} \\ =& u^{2} \cdot 0 + 2 \cdot 0 + v^{2} dt \\ =& v^{2} dt \end{align*} $$ is obtained.
Explanation
The Itô formula is also known as the Itô lemma or Itô chain rule and is a crucial theorem used throughout stochastic differential equations. Because it frequently appears in almost all calculations, it certainly merits being called a chain rule.
The proof is omitted here, but simply applies the multivariable Taylor theorem and disregards higher-order terms.
Example
Integrating a Wiener process by a Wiener process is intuitively difficult to understand. Formally, one might naturally expect a result like $\displaystyle \int_{0}^{t} W_{s} d W_{s} = {{ 1 } \over { 2 }} W_{t}^{2}$, similar to Riemann integration, but let’s calculate and see if this is the case.
Before using the Itô formula, set the given Itô process to $u = 0$, $v = 1$ such that $$ d X_{t} = 0 dt + 1 d W_{t} $$ then $X_{t} = W_{t}$. Here, if we let $\displaystyle Y_{t} := V \left( t , X_{t} \right) = {{ X_{t}^{2} } \over { 2 }}$, $$ \begin{align*} V_{t} =& {{ \partial } \over { \partial t }} \left( {{ 1 } \over { 2 }} W_{t}^{2} \right) = 0 \\ V_{x} =& {{ \partial } \over { \partial W_{t} }} \left( {{ 1 } \over { 2 }} W_{t}^{2} \right) = W_{t} \\ V_{xx} =& {{ \partial^{2} } \over { \partial W_{t}^{2} }} \left( {{ 1 } \over { 2 }} W_{t}^{2} \right) = {{ \partial } \over { \partial W_{t} }} W_{t} = 1 \end{align*} $$ then from $u = 0$, $v = 1$, $$ \begin{align*} d \left( {{ W_{t}^{2} } \over { 2 }} \right) =& \left( V_{t} + V_{x} u + {{ 1 } \over { 2 }} V_{xx} v^{2} \right) dt + V_{x} v d W_{t} \\ =& \left( 0 + W_{t} \cdot 0 + {{ 1 } \over { 2 }} \cdot 1 \cdot 1^{2} \right) dt + W_{t} \cdot 1 \cdot d W_{t} \\ =& {{ 1 } \over { 2 }} dt + W_{t} d W_{t} \end{align*} $$ converting the differential form to the integral form yields $$ {{ W_{t}^{2} } \over { 2 }} = {{ 1 } \over { 2 }} t + \int_{0}^{t} W_{s} d W_{s} $$ Summarizing gives the following. $$ \int_{0}^{t} W_{s} d W_{s} = {{ 1 } \over { 2 }} \left( W_{t}^{2} - t \right) $$ At first glance, the presence of the $1$th term $-t /2$, which was not in Riemann integration, might seem messy and inconvenient. However, considering taking the expectation, $$ t = \text{Var} \left( W_{t} \right) = E \left( W_{t}^{2} \right) - 0^{2} $$ thus it disappears neatly as $$ E \left( \int_{0}^{t} W_{s} d W_{s} \right) = {{ 1 } \over { 2 }} \left( t - t \right) = 0 $$. The $1$th term is not just garbage generated due to a difference in calculation methods but has its own meaningful existence. If one concurs that the expectation when integrating a Wiener process by a Wiener process should be $0$, then one can intuitively accept the above conclusion.
Stochastic Integration 2
Let $a < b$ and $c$ be a constant and let $t > 0$.
$$ \begin{align*} \int_{0}^{t} d W_{s} =& W_{t} \\ \int_{a}^{b} c d W_{s} =& c \left[ W_{b} - W_{a} \right] \end{align*} $$
The above two cases yield the same results as ordinary Riemann integration, but the following yields a unique result only seen in Itô integration.
$$ \begin{align*} \int_{0}^{t} W_{s} d W_{s} =& {{ 1 } \over { 2 }} W_{t}^{2} - {{ 1 } \over { 2 }} t \\ \int_{a}^{b} W_{s} d W_{s} =& {{ 1 } \over { 2 }} \left[ W_{b}^{2} - W_{a}^{2} \right] - {{ 1 } \over { 2 }} (b-a) \\ \int_{0}^{t} s d W_{s} =& t W_{t} - \int_{0}^{t} W_{s} ds = (t-1) W_{t} \\ \int_{0}^{t} W_{s}^{2} d W_{s} =& {{ 1 } \over { 3 }} W_{t}^{3} - \int_{0}^{t} W_{s} ds \\ \int_{0}^{t} e^{W_{s}} d W_{s} =& e^{W_{t}} - 1 - {{ 1 } \over { 2 }} \int_{0}^{t} e^{W_{s}} ds \\ \int_{0}^{t} W_{t} e^{W_{s}} d W_{s} =& 1 + W_{t} e^{W_{t}} - e^{W_{t}} - {{ 1 } \over { 2 }} \int_{0}^{t} e^{W_{s}} \left( 1 + W_{s} \right) d W_{s} \\ \int_{0}^{t} s W_{s} d W_{s} =& {{ t } \over { 2 }} \left( W_{t}^{2} - {{ t } \over { 2 }} \right) - {{ 1 } \over { 2 }} \int_{0}^{t} W_{s}^{2} ds \\ \int_{0}^{t} \left( W_{s}^{2} - s \right) d W_{s} =& {{ 1 } \over { 3 }} W_{t}^{3} - t W_{t} \\ \int_{0}^{t} e^{-s/2 + W_{s}} d W_{s} =& e^{-t/2 + W_{t}} - 1 \\ \int_{0}^{t} \sin W_{s} d W_{s} =& 1 - \cos W_{t} - {{ 1 } \over { 2 }} \int_{0}^{t} \cos W_{s} ds \\ \int_{0}^{t} \cos W_{s} d W_{s} =& \sin W_{t} + {{ 1 } \over { 2 }} \int_{0}^{t} \sin W_{s} ds \end{align*} $$
Especially regarding the expectation and variance, the following equations are known.
$$ \begin{align*} E \left( \int_{0}^{t} d W_{s} \right) =& 0 \\ E \left( \int_{0}^{t} W_{s} d W_{s} \right) =& 0 \\ \text{Var} \left( \int_{0}^{t} W_{s} d W_{s} \right) =& {{ t^{2} } \over { 2 }} \end{align*} $$