Café Math : Brownian Motion (Part 1)
1, 1, 2, 2, 1, 8, 6, 7, 14, 27, 26, 80, 133, 170, 348, 765, 1002, 2176, 4682, ...   (A121350)
Math Blog
Phd Thesis
Financial Maths
About me

Brownian Motion (Part 1)

Mon, 6 Feb 2012 00:44:35 GMT

[Download R code]

1. Foreword

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.

2. Basic Properties

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,

  1. It passes through the origin : $W_0 = 0$.
  2. 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 $0 \le t_1 \le t_2 \le t_3 \le t_4$.
  3. 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 $\sigma\sqrt{t_2 - t_1}$.
  4. 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.

  1. Symmetry : $$ Y_t = -W_t $$
  2. Scaling : $$ Y_t = s \,W_{t/s^2} $$
  3. Translation : $$ Y_t = W_{t + T} - W_T $$
  4. Reflexion principle : $$ Y_t = \begin{cases} W_t & t \le T \\ 2\,W_T - W_t & t \ge T \end{cases} $$
  5. 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 $\sigma$. $$ Y_t = \begin{cases} W_t & t \le T \\ W_T + W'_{t-T} & t \ge T \end{cases} $$

3. Numerical Simulation

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.


Samuel VIDAL  posted 2012-05-21 00:50:33

One cool video,

Smithb937  posted 2014-08-28 05:03:12

Greetings! Very useful advice within this article! It is the little changes that make the most important changes. Many thanks for sharing! dceccegdcdefadkg

Pharme3  posted 2014-08-28 18:45:00

Very nice site! cheap goods

Pharmd725  posted 2014-08-28 18:48:41

Very nice site! [url=]cheap goods[/url]

Pharmk643  posted 2014-08-28 18:54:05

Very nice site! cheap goods

Pharme961  posted 2014-08-29 00:28:34

Very nice site! cheap goods

Pharmg230  posted 2014-08-29 00:29:04

Very nice site! [url=]cheap goods[/url]

Pharmk960  posted 2014-08-29 00:29:46

Very nice site! cheap goods

Pharmd923  posted 2014-08-29 00:30:12

Very nice site!

Pharma814  posted 2014-08-30 06:48:48

Very nice site! cheap goods

Pharma93  posted 2014-08-30 06:49:18

Very nice site! cheap goods

Pharme586  posted 2014-08-30 06:49:54

Very nice site!

Pharmd701  posted 2014-08-31 13:12:38

Very nice site! [url=]cheap goods[/url]

Pharme745  posted 2014-08-31 13:15:36

Very nice site! cheap goods

Pharmd889  posted 2014-08-31 13:16:09

Very nice site!

Pharme961  posted 2014-09-01 19:27:25

Very nice site! cheap goods

Pharmc135  posted 2014-09-01 19:29:59

Very nice site! [url=]cheap goods[/url]

Pharmf761  posted 2014-09-01 19:31:34

Very nice site! cheap goods