Category Archives: Stochastic Calculus

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

What is a Stochastic Differential Equation?

Consider the population growth model
\begin{equation}\label{eq:popgrowth}\frac{dN}{dt}=a(t)N(t),\ N(0)=N_0\end{equation}
where $N(t)$ is the size of a population at time $t$ and $a(t)$ is the relativive growth rate at time $t$. If $a(t)$ is completely known, one can easily solve \eqref{eq:popgrowth}. In fact, the solution would be $N(t)=N_0\exp\left(\int_0^t a(t)dt\right)$. Now suppose that $a(t)$ is not completely known but it can be written as $a(t)=r(t)+\mbox{noise}$. We do not know the exact behavior of noise but only its probability distribution. Such a case equations like \eqref{eq:popgrowth} is called a stochastic differential equation. More genrally, a stochastic differential equation can be written as
\begin{equation}\label{eq:sd}\frac{dX}{dt}=b(X(t))+B(X(t))\xi(t)\ (t>0),\ X(0)=x_0,\end{equation}
where $b: \mathbb{R}^n\longrightarrow\mathbb{R}^n$ is a smooth vector field and $X: [0,\infty)\longrightarrow\mathbb{R}^n$, $B: \mathbb{R}^n\longrightarrow\mathbb{M}^{n\times m}$ and $\xi(t)$ is an $m$-dimensional white noise. If $m=n$, $x_0=0$, $b=0$ and $B=I$, then \eqref{eq:sd} turns into
\begin{equation}\label{eq:wiener}\frac{dX}{dt}=\xi(t),\ X(0)=0\end{equation}
The solution of \eqref{eq:wiener} is denoted by $W(t)$ and is called the $n$-dimensional Wiener process or Brownian motion. In other words, white noise $\xi(t)$ is the time derivative of the Wiener process. Replace $\xi(t)$ in \eqref{eq:sd} by $\frac{W(t)}{dt}$ and divide the resulting equation by $dt$. Then we obtain
\begin{equation}\label{eq:sd2}dX(t)=b(X(t))dt+B(X(t))dW(t),\ X(0)=x_0\end{equation}
The stochastic differential equation \eqref{eq:sd2} is solved symbolically as
\begin{equation}\label{eq:sdsol}X(t)=x_0+\int_0^tb(X(s))ds+\int_0^tb(X(s))dW(s)\end{equation}
for all $t>0$. In order to make sense of $X(t)$ in \eqref{eq:sdsol} we will have to know what $W(t)$ is and what the integral $\int_0^tb(X(s))dW(s)$, which is called a stochastic integral, means.

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