Café Math : Itô Calculus (Part 1)

Today I want to talk about something very useful called Itô calculus. The stochastic differential calculus invented by Itô is a way to extend the ordinary differential calculus by extending the rules a bit, so that it can handle computations involving stochastic processes. Of course Itô calculus is not applicable to every stochastic processes, in the same way that ordinary differential calculus is not applicable to every functions. Some technical assumptions have to be fulfilled for it to apply. The variety of cases that Itô calculus permits is already wide enough so as to make it a weapon of choice.

As one can find all the technical details in any good textbook on the subject, I'll simplify the presentation by omitting a lot of the formal justifications. My point is that one can already get a feel of Itô calculus by playing with the rules of computation and by doing some computer experiments.

Let $B$ be a standard brownian motion and $X$ be a stochastic process whose randomness comes only from $B$. For instance $X$ can be a function of $B$ and some other deterministic quantities like time $t$.

Definition 1. An Itô stochastic differential equation (or Itô equation for short) is an expression of the following form, $$ dX = \mu dt + \sigma dB $$ where $dB$ is the differential of a standard brownian motion $B$.

We see that an Itô equation decomposes the differential of $X$ in two components, its *deterministic* part $\mu dt$ and its *stochastic* part $\sigma dB$.
The variable $dB$ is the differential of $B$, it is called by physicists the driving noise of the equation.
The quantities $\mu$ and $\sigma$ are called the drift and the dispersion of the process $X$.

There are basic rules of calculus that are very simple yet very useful to remember. \begin{align*} dt^2 &= 0 & dB^2 &= dt & dB\,dt &= 0 \end{align*} Of course, more rules are to be added when there is more than one brownian motion in the equation but let's stick to the simple dimension one case for now.

Now let's talk about an analogue of the pullback formula in the context of stochastic differential calculus. The question is : What is the stochastic differential equation that satisfies a process $Y$ related to another stochastic process $X$ by a determinist relation $Y = f(X,t)$ when you know a stochastic equation for $X$ ?

So, given a stochastic processes $X$ described by an Itô equation, $$ dX = \mu \, dt + \sigma \, dB $$ and a process $Y$ subject to the relation, $$ Y = f(X,t) $$ we ask for an expression of the form, $$ dY = \mu' \, dt + \sigma' \, dB $$

Whenever $f$ is twice differentiable with respect to $X$ and once differentiable with respect to $t$, the answer is given by the Itô formula, $$ dY = \left[ \frac{\partial}{\partial t} Y + \mu \frac{\partial}{\partial X} Y + \tfrac{1}{2} \sigma^2 \frac{\partial^2}{\partial X^2}Y \right] dt + \sigma \frac{\partial}{\partial X}Y dB $$

This formula is a direct consequence of the basic multiplication table. Let's see. First, using the Taylor formula we get, $$ dY = \frac{\partial}{\partial t} Y \,dt + \frac{\partial}{\partial X} Y \,dX + \tfrac{1}{2}\frac{\partial^2}{\partial X^2} Y \,dX^2 $$ and now using the expression of $dX$ and the basic multiplication table, $$ \begin{aligned} dX &= \mu \, dt + \sigma\, dB \\ dX^2 &= (\mu \, dt + \sigma \, dB)^2 \\ &= \mu^2 \, dt^2 + 2\,\mu\sigma\, dt dB + \sigma^2 \, dB^2\\ &= \sigma^2 \, dt \end{aligned} $$ we are ready to replace the value of $dX$ and $dX^2$ in the Taylor expansion by the above expressions. $$ \begin{aligned} dY &= \frac{\partial}{\partial t} Y \,dt + \left[\mu \frac{\partial}{\partial X} Y\, dt +\sigma \frac{\partial}{\partial X}Y dB \right]+ \tfrac{1}{2} \,\sigma^2\frac{\partial^2}{\partial X^2} Y \, dt \\ &= \left[ \frac{\partial}{\partial t} Y + \mu \frac{\partial}{\partial X} Y + \tfrac{1}{2} \sigma^2 \frac{\partial^2}{\partial X^2}Y \right] dt + \sigma \frac{\partial}{\partial X}Y dB \end{aligned} $$

Now you can ask : what about the terms $dt \, dX$, $dX^3$, etc... why not consider them in the Taylor expansion ? Well by the basic multiplication rules, those terms vanish. $$ \begin{aligned} dt \, dX &= \mu \, dt^2 + \sigma \, dt \, dB = 0 \\ dX^3 &= \mu^3 \, dt^3 + 3\, \mu^2 \sigma \, dt^2 \,dB + 3\, \mu \sigma^2 \, dt \, dB^2 + \sigma^3 \, dB^3 = 0 \\ \dots \end{aligned} $$

Let's look at some examples.

Example (*Arithmetic brownian motion*).
An *arithmetic brownian motion* (with drift), is a stochastic process $X$ subject to a differential equation of the form,
$$
dX = \mu \, dt + \sigma \, dB
$$
where $B$ is a standard brownian motion, and both the *drift coefficient* $\mu$ and the *diffusion coefficient* $\sigma$ are constants.
The solution to this differential equation is simply,
$$
X_t = X_0 + \mu \, t + \sigma \, B_t
$$
can you prove that ?

Example (*Geometric brownian motion*).
A *geometric brownian motion* is a stocastic process $S$ subject to a differential equation of the form,
$$
dS = \mu \, S \, dt + \sigma \, S \, dB
$$
where again, $B$ is a standard brownian motion, and both the *drift coefficient* $\mu$ and the *diffusion coefficient* $\sigma$ are constants.

Solving for $S$ we get, $$ S = S_0\,e^{(\mu - \tfrac{1}{2}\sigma^2)t + \sigma\, B_t} $$

So $S$ is, up to a constant factor, the exponential of an arithmetic brownian motion with drift $\mu - \tfrac{1}{2}\sigma^2$ and dispersion $\sigma$.

In another article we'll talk about numerical simulations of stochastic processes described by Itô stochastic equations, and we'll do some computer experiments, but not today.

I never studied stochastic calculus before and it turns out that this is the lingua franca of finance. That's true. I am developing a Java software library or API (application programming interface) in this area to be used in a web application. I love doing numerical modeling in Finance (Stochastic Differential Calculus), since it captured all the methods used in other scientific disciplines. For example, topics that are applicable in numerical Finance comes from Differential Calculus, Statistics, Physics (well known one is the Black-Scholes model which is solved using the heat equation of thermo-dynamics), Signal Processing , Feedback Control Theory & Systems Dynamics, and more. Interesting field.PS : I never formally studied finance, just being self-taught, since I am familiar with the maths already (which I specialize in scientific computing) my transition to learning computational finance was no hassle.