SDE: Itô's Formula

 Let us consider the 1-dimensional case ($n=1$) of the stochastic differential equation (4) in here
$$dX=b(X)dt+dW \tag{1}$$
with $X(0)=0$.
Let $u: \mathbb{R}\longrightarrow\mathbb{R}$ be a smooth function and $Y(t)=u(X(t))$ ($t\geq 0$). What we learned in calculus (the chain rule) would dictate us that $dY$ is
$$dY=u'dX=u'bdt+u'dW,$$
where $'=\frac{d}{dx}$. It may come to you as a surprise to hear this but this is not correct in case of stochastic processes. First, by Taylor series expansion, we obtain
\begin{align*}
\Delta Y&=u(X+\Delta X)-u(X)\\
&=u(X)+u'(X)\Delta X+\frac{u''(X)}{2!}(\Delta X)^2+\cdots -u(X)\\
&=u'(X)\Delta X+\frac{u''(X)}{2!}(\Delta X)^2+\cdots
\end{align*}
and thus we have
\begin{align*}
dY&=u'dX+\frac{1}{2}u^{\prime\prime}(dX)^2+\cdots\\
&=u'(bdt+dW)+\frac{1}{2}u^{\prime\prime}(bdt+dW)^2+\cdots
\end{align*}
Now, we introduce the following striking formula
$$(dW)^2=dt \tag{2}$$
The proof of (2) is beyond the scope of this note and so it won't be discussed here. However it can be found, for example, in [1]. Using (2) $dY$ can be written as
$$dY=\left(u'b+\frac{1}{2}u^{\prime\prime}\right)dt+u'dW+\cdots$$
The terms beyond $u'dW$ are of order $(dt)^{\frac{3}{2}}$ and higher. Neglecting these terms, we have
$$dY=\left(u'b+\frac{1}{2}u^{\prime\prime}\right)dt+u'dW \tag{3}$$
(3) is the stochastic differential equation satisfied by $Y(t)$ and it is called the Itô's Formula named after a Japanese mathematician Kiyosi Itô.

Example. Let us consider the stochastic differential equation
$$dY=YdW,\ Y(0)=1 \tag{4}$$
Comparing (3) and (4), we obtain
\begin{align*}
u'b+\frac{1}{2}u^{\prime\prime}&=0 \tag{5}\\u'&=u \tag{6}
\end{align*}
The equation (6) along with the initial condition $Y(0)=1$ results in $u(X(t))=e^{X(t)}$. Using this $u$ with equation (5) we get $b=-\frac{1}{2}$ and so the equation (1) becomes
$$dX=-\frac{1}{2}dt+dW$$
in which case $X(t)=-\frac{1}{2}t+W(t)$. Hence, we find $Y(t)$ as
$$Y(t)=e^{-\frac{1}{2}t+W(t)}$$

Example. Let $P(t)$ denote the price of a stock at time $t\geq 0$. A standard model assumes that the relative change of price $\frac{dP}{P}$ evolves according to the stochastic differential equation
$$\frac{dP}{P}=\mu dt+\sigma dW \tag{7}$$
where $\mu>0$ and $\sigma$ are constants called the drift and the volatility of the stock, respectively. Again using Itô's formula similarly to what we did in the preceding example, we find the price function $P(t)$ which is the solution of
$$dP=\mu Pdt+\sigma PdW,\ P(0)=p_0$$
as
$$P(t)=p_0\exp\left[\left(\mu-\frac{1}{2}\sigma^2\right)\right]t+\sigma W(t).$$

Example. In this example, we solve the stochastic population growth model
\begin{align*}
\frac{dN}{dt}&=(r(t)+\xi(t))N(t) \tag{8}\\
&=r(t)N(t)+\xi(t)N(t)
\end{align*}
with $N(0)=N_0$. Here, $r(t)$ is a known function (this $r(t)$ may be considered as the relative growth rate at time $t$ in the deterministic population growth model) and $\xi(t)=\frac{dW}{dt}$ is a white noise. Let $N(t)=u(X(t))$. Then
$$dN=r(t)udt+udW \tag{9}$$
Comparing (3) and (9), we obtain
\begin{align*}
u'b+\frac{1}{2}u''&=r(t)u\\
u'&=u
\end{align*}
$u(t)=N_0e^{X(t)}$ and hence we find $b=r(t)-\frac{1}{2}$. From the equation (1), we have
$$dX=\left(r(t)-\frac{1}{2}\right)dt+dW$$
whose solution $X(t)$ is given by
$$X(t)=\int_0^t r(s)ds-\frac{t}{2}+W(t)$$
Therefore,
$$N(t)=N_0\exp\left[\int_0^t r(s)ds-\frac{t}{2}+W(t)\right]$$

References:

1. Bernt Øksendal, Stochastic Differential Equations, An Introduction with Applications, 5th Edition, Springer, 2000