# Chapter 9

Discussion in 'CM2' started by Chirag Wadhwa, May 19, 2022.

1. ### Chirag WadhwaKeen member

Can anyone explain what filtration and natural filtration imply and represents?

What do drift coefficients and diffusion mean in terms of Brownian Motion?

In the proof of non-differentiability of Sample paths, can anyone explain the case for t>s?
How (W_t-W_s)/(t-s)-dW_t/dt~N(-dW_t/dt,1/t-s) and the explanation afterwards?

2. ### CapitalActuaryVery Active Member

Unless the acted notes have changed a lot since I took the exams, I seem to recall there being quite a good explanation of filtrations in there. A natural filtration is just something that records the history of the stochastic process. Note when I did the exams, I got the distinct impression you don't really need to understand these things properly as you never get asked questions about them in the exams. Perhaps this has changed - but I suspect not because I don't think most student actuaries are comfortable enough with measure theory to really understand all this stuff. I wouldn't worry about it.

A Brownian motion is a continuous random walk that wiggles along going up and down over time. If the drift is positive, then on average over time the Brownian motion wiggles upwards. It might have long periods where it wiggles down, but in the long run it will wiggle its way up. The diffusion just controls how big the wiggles tend to be, but not which the average long run direction in which the process wiggles.

The case for t>s shows that the thing you want to be smaller than an arbitrary epsilon when t-s approaches 0 has a normal distribution with variance that gets massive, infinite in fact, as t-s approaches 0. The probability of a normally distributed random variable with essentially infinite variance having smaller magnitude than any epsilon is 0. That's what the notes mean when they say the expression holds 'almost surely' (which is a technical term meaning "with probability 1").

I think you asked somewhere else on the forum a similar question about how some expression involving W terms had normal distribution with certain parameters. Have a look at my answer there to see if it clarifies things - and also check again you understand the definition of a Wiener process / Brownian motion and what that has to do with the normal distribution.

If you'd like more explanation it would be helpful to understand which specific bits you don't understand, as I am not really sure whether in attempting to provide explanations to broader questions I am just repeating explanations which are already there in the acted notes or not.

Last edited: May 22, 2022