Talk:Brownian motion

I think that the mathematical section should make reference to the Central limit theorem, which explains (as far as I know) why the position of a particle at a time t can be considered as normally distributed random variable. Psychofox 01:37, Mar 21, 2005 (UTC)

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I think the entry would benefit from an example such as:

Informal Discussion

Suppose a computer programmer is assigned the task of simulating Brownian motion in 1-dimension. He writes a program where a particle starts at x = 0 on the x-axis at time t = 0. Every second he picks random numbers to decide whether the particle makes a jump of +1 or -1. Some sample paths of the particle are plotted and show to the boss.

The boss is delighted. But, being a boss, he says "This is fine, but I want to you to fix your program so the user can select the time interval for the jumps to 1/2 second or 1/3 second or whatever he wants. Some users may want a higher resolution simulation."

So the programmer modifies his program so the user can input the time between jumps.He makes some sample plots using a time interval of 1/10 second and is dismayed to find that the look much more erratic that his original plots, make at intervals of 1 second.

Upon reflection, he understands why. In the original program the position of the particle after, say, 8 seconds was the result of 8 jumps and in the new plots it is the result of (8)(10) = 80 jumps.

He thinks to himself: "If the user inputs a time interval of 1/n, I will change the jump size to 1/n also. That way after 8 seconds the total jumps will be 8n but the max distance the particle can move will be (8n)(1/n) = 8, which is the same as in my original program that used a time interval of 1 second and a jump size of 1."

He makes this change and runs some test plots using a time interval of 1/10. He is dismayed to find that the particle seems to be less variable than it was using the 1 second time interval. He tries using time intervals of 1/100 and 1/1000 seconds and finds that the smaller the time interval, the more the particle tends to stay where it is.

Then he does a mathematical analysis to un-confuse himself. Suppose the time interval is dt = 1/n and the jump size is dx. After 8 seconds there are 8/dt = 8n steps, each is of length +dx or -dx. So the position of the particle at time 8 is the sum of 8n = 8/dt independent random variables, each with value +dx or -dx. The sum of independent random variables can be approximated as a normal distribution, even if the variables themselves are not normally distributed. Let N(m,s) be the normal distribution with mean m and standard deviation s. The approximating normal will have m = the sum of the means of jumps. Each jump, as a single random variable, has mean = 0, so m = 0. The value of s is given by s^2 = the sum of the variances of the jumps. A single jump is a random variable which has value dx with probability 0.5 and value -dx with probability 0.5. The calculation for its variance is (0.5)(dx - 0)^2 + 0.5 (-dx - 0)^2 = dx^2. Since there are 8n = 8/dt such random variables in the sum for s^2, we have that s^2 = 8n (dx^2) = (8/dt)(dx^2) = 8 dx^2/ dt.

It is now clear that if we set the jump size dx to be 1/n when the time interval is dt = 1/n. Then s^2 is 8 dx^2/dt = 8(1/n)^2/(1/n) = 8/n. So as n gets larger (making dt smaller) the standard deviation s, gets smaller and smaller. This explains why the position of the particle after 8 seconds tends to be near x = 0. The probability distribution for its location is concentrated near the mean m = 0.

But how can he fix his program so the variability of the particle approximates what it was in the orginal plots he showed to the boss? In those plots s^2 = 8 1^2/1 = 8. We need to find a dx so 8 dx^2/dt = 8. Solving for dx gives dx^2 = dt and dx = sqrt(dt).

So when the user inputs a time step of (1/n) = dt, we should use a jump size of sqrt(1/n) = sqrt(dt).

He implements this procedure and the resulting plots make the boss happy.

The above story indicates that there are various intellectual problems in going from the verbal description Brownian motion as "the random motion of a small particle" to a precise mathematical theory.

One problem is to determine whether the programmer's algorithm is, in some sense, approximating a process that takes places in continuous time rather than in discrete time steps. This can be established if we precisely define what it means for a sequence of stochastic processes to converge to limiting stochastic process. (This can be done.)

Another question is whether the limit of this sequence of processes is the only kind of continuous random motion that is possible. Apparently it is not. For example, we could allow the user to input a constant factor k and multiply all the jumps of the particle by k. So a better question is whether the limiting process or some constant multiple of it gives describes all possible types of continuous random motions. They don't. However these do constitute an important set of such process that are widely used to analyze phenomena is physics and economics.

(Perhaps we should also mention Wiener and Kholmogorov. I notice that "limit o f a sequence of random variables" is not yet treated in the Wikipedia.)

The caption

The caption to the first picture says the variance is 2. What does that mean when we're talking about a vector-valued random variable, rather than scalar-valued? Often one speaks of a covariance matrix, or of a "variance" that is that matrix or is the associated linear transformation. Michael Hardy 23:42, 23 Jan 2005 (UTC)

Right, I changed the caption to make that clear. Paul Reiser 05:27, 24 Jan 2005 (UTC)

Merge with Wiener process?

I've heard that Wiener process is simply another name for Brownian motion. Is this true? If so should the articles not be merged? reetep 21:32, 3 Jun 2005 (UTC)

Within mathematics, it is true. In the physical sciences, Brownian motion is the erratic motion of tiny particles suspended in a fluid. The Wiener process is a mathematical model that has been proposed to model that and various other phenomena. Whether the Wiener process adequately models Brownian motion is a question to be decided in part by empirical observation. That they are in some sense the same is hardly an a priori truth. Michael Hardy 21:37, 3 Jun 2005 (UTC)