Café Math : Brownian Motion (Part 1)

Hi, today I want to talk about the important concept of brownian motion. In the first part, I will recall what are brownian motions and what are their basic properties. After that, we'll look at practical ways to simulate them.

Brownian motions constitute a particular class of continuous time stochastic processes. They basically are families of random variables parameterized by the non-negative reals.

Definition 1.
A *brownian motion* $W_t$ with *dispersion coefficient* $\sigma \ge 0$, also denoted $W(t)$ or simply $W$, is a stochastic process such that,

- It passes through the origin : $W_0 = 0$.
- It has independent increments on non-overlapping intervals : The two random variables, $$ \Delta_1 = W_{t_2} - W_{t_1} \qquad\text{ and }\qquad \Delta_2 = W_{t_4} - W_{t_3} $$ are independent as soon as $0 \le t_1 \le t_2 \le t_3 \le t_4$.
- Increments are normally distributed : For all $t_1$ and $t_2$ with $0 \le t_1 \le t_2$, the random variable $W_{t_2} - W_{t_1}$ has normal distribution, with mean $0$ and standard deviation $\sigma\sqrt{t_2 - t_1}$.
- With probability $1$, $W_t$ is continuous as a function of $t$.

In other words, the variance of the increments are proportional to the length of the corresponding interval. The proportionality coefficient is the square $\sigma^2$ of the dispersion. A brownian motion is *qualified standard* when $\sigma = 1$.

The dynamic of brownian motions is both rich and simple. Rich in that it models a lot of natural phenomenon of stochastic nature. Simple in that it is subject to some very nice symmetries. For instance, each of the following relation defines a brownian motion $Y$ from a given one $W$, both of the same dispersion.

- Symmetry : $$ Y_t = -W_t $$
- Scaling : $$ Y_t = s \,W_{t/s^2} $$
- Translation : $$ Y_t = W_{t + T} - W_T $$
- Reflexion principle : $$ Y_t = \begin{cases} W_t & t \le T \\ 2\,W_T - W_t & t \ge T \end{cases} $$
- Time inversion : $$ Y_t = \begin{cases} 0 & t = 0\\ t\, W_{1/t} & t > 0 \end{cases} $$

Each of those symmetries is in fact a theorem worth proving. We invite the interested reader to try to prove them using only the properties of the definition.

One more thing, if $W$ and $W'$ are two independent brownian motions having the same dispersion $\sigma$, then one can piece them together at a particular point of time $T$ to produce another brownian motion $Y$ of dispersion $\sigma$. $$ Y_t = \begin{cases} W_t & t \le T \\ W_T + W'_{t-T} & t \ge T \end{cases} $$

Suppose first that we want to simulate the values taken by a brownian motion $W$ at regularly spaced time intervals. We call $\Delta t$ the distance between two successive time intervals. Then for $k = 0, 1, ...$ we have $t_k = k \, \Delta t$ and $$ W_{t_{k+1}} = W_{t_k} + \sigma\,\sqrt{\Delta t}\, X_k $$ where the $X_k \sim \mathcal{N}(0,1)$ are independent standard gaussian variables. Standard here means, of zero mean and unit variance.

We've just approximated a brownian motion by a discrete stochastic process (i.e. a sequence of random variables $(U_k)_{k \ge 0}$ with $U_k = W_{t_k}$. But the converse is also possible and useful.

Consider a discreet stochastic process $(U_k)_{k \ge 0}$ whose increments are independents and identically distributed of zero mean and variance $\sigma$.
Then, given a time scale $\Delta t$, a brownian motion of dispersion $$\frac{\sigma}{\sqrt{\Delta t}}$$ is a very convincing approximation of $U_k$,
especially if $\Delta t$ is considered small. However, in the case where $\Delta t$ isn't small, the approximation can be considered very coarse,
except of course when the increments are normally distributed where the approximation is *perfect*.