Stochastic Differential Equations 2: A Taste of Itô’s Formula

Let us consider the 1-dimensional case ($n=1$) of the Stochastic Equation (4) from the last post
\begin{equation}\label{eq:sd3}dX=b(X)dt+dW\end{equation} 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. First by Taylor series expansion we obtain
\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
\begin{equation}\label{eq:wiener2}(dW)^2=dt\end{equation}
The proof of \eqref{eq:wiener2} is beyond the scope of this notes and so it won’t be given now or ever. However it can be found, for example, in [2]. Using \eqref{eq:wiener2} $dY$ 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
\begin{equation}\label{eq:sd4}dY=\left(u’b+\frac{1}{2}u^{\prime\prime}\right)dt+u’dW\end{equation}
\eqref{eq:sd4} 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
\begin{equation}\label{eq:sd5}dY=YdW,\ Y(0)=1\end{equation}
Comparing \eqref{eq:sd4} and \eqref{eq:sd5}, we obtain
\begin{align}\label{eq:sd5a}
u’b+\frac{1}{2}u^{\prime\prime}&=0\\\label{eq:sd5b}u’&=u\end{align}
The equation \eqref{eq:sd5b} along with the initial condition $Y(0)=1$ results $u(X(t))=e^{X(t)}$. Using this $u$ with equation \eqref{eq:sd5a} we get $b=-\frac{1}{2}$ and so the equation \eqref{eq:sd3} 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
\begin{equation}\label{eq:relprice}\frac{dP}{P}=\mu dt+\sigma dW\end{equation}
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 previous example, we find the price function $P(t)$ which is the solution of
$$dP=\mu Pdt+\sigma PdW,\ P(0)=p_0$$
as
$$P(t)=p_0\exp\left[\left(\mu-\frac{1}{2}\sigma^2\right)\right]t+\sigma W(t).$$

References:

1. Lawrence C. Evans, An Introduction to Stochastic Differential Equations, Lecture Notes

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

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