Brownian motion #1 (basic properties)

okay the purpose of today's video is to introduce you to the concept of standard Brownian motion okay so let's start with basic properties of standard Brownian motion standard Brownian motion here denoted as W indexed with time so W at time 0 equals 0 what this means is that our Brownian motion at time equals 0 takes the value of 0 and this is best explained looking at the graph here I've got here a realization of my Brownian motion path the Brownian motion decided to take this path and this is just one realization of Brownian motion ok but one thing that all the realizations of Brownian motions will have is that and it will take a value of 0 at time equals 0 so let's denote the W at time 0 equals 0 ok so here on this axis we've got time on the x-axis a good time and on the y-axis we've got W which is the values of our Brownian motion now clearly this property was pretty much self explanatory let's have a look at the other properties property number 2 Brownian motion is normally distributed with mean 0 and variance T and Brownian motion increment W small t minus W small s is also normally distributed with mean 0 and variance t minus s ok so what does it mean that the Brownian motion is normally distributed with mean 0 and variance T so say I'm standing here at time equals 0 and I want to know what my Brownian motion is or will be at time say time equals T equals 2 ok so I see here that this single realization of Brownian motion happened to take a value of 0.4 okay but this is just one real realization on the other realization Brownian motion can end up here here the there there it can pretty much end up all over the shop ie it can take a range of values the good news however is that I will know what the expected value okay this is the distribution of Brownian motion at time equal to okay but the good news I will know what the center of the distribution is ie what what the expected value of this distribution is and and this will be expected value of W at time two equals zero K it will always be zero whether I look at the Brownian motion here here or here the expectation of bromium motion at all these points is zero so not only the mean or our expect expected value that the value of the Brownian motion at any time will be zero but also we are told that it will be normally distributed that's why I'm here I graphed like a normal distribution okay the peak of the distribution is centered at zero meaning that my Brownian motion will be distributed as a normal variable with expected value 0 and variance T ok now crucially it says T here what does it mean well if I look at the distribution of my Brownian motion say at this point here then I will see that this distribution is also centered at 0 but the variance of this distribution say at time equals T equals 1 is 1 ok now the variance of this distribution Brownian motion at this point will be 2 okay so my variance will basically so if I have here just drawing on the graph my variance will increase proportionally to time so this is variance I will denote here Sigma squared is proportional is of order T ok I think I already spent too much time with this to illustrate such a simple concept I only spoke about WT but the there is a related concept ie the Brownian motion increment which is the difference between two Brownian motions okay at some one at time T the other one time s and it turns out that the difference between the two Brownian motion is also normally distributed and the fineness of the Brownian motion increments ie the difference between two Brownian motion is just t minus s T stands for time and s stands for time is just the difference in time between the two bro between the measurements of our Brownian motions okay so what this means if if I look at my Brownian motion somewhere at this point here and somewhere at this point here so the expected increment so this is time T this is time s the expectation of the difference of these two Brownian motion VT minus V s equals 0 and the variance of this difference equals t minus s yet again it's proportional violence proportional to time will show these properties more formally later on and one by one but let's see let's continue let's discuss other properties of Brownian motion the process WT has stationary and independent increment okay so what does it mean that the Brownian motion has stationary increments well if I look at my Brownian motion at time 0 and say my Brownian motion at time 1 and then I looked at my display sprunjer motion I move the Brownian motion increment further in time by a constant a ok so this is going be to W 0 plus a and W 1 plus a and what this means is that the distribution of this increment here okay will be exactly the same as the distribution of this increment here ie day we'll both be normal with expected value of 0 and variance of T 1 minus T 0 so it doesn't matter where you look at the Brownian motion it will always have the same properties okay its expectation so we're talking about increments the incremental expectation will be always 0 and the variance will be just the difference difference in time so to sum up independently where you look at the Brownian motion whether you look at Brownian motion here in this time interval or here it will always have the same property in probabilistic sense okay will always have an expectation of 0 and variance will be proportional to time another property is that increments save W 1 less than p 0 and W 1 plus a let's W 0 plus a what independence says is that whatever however I moved in this time interval will have absolutely no bearing on how I'm going to move in this time interval okay I'm going to clear this graph let's see some other properties of Brownian motion it will briefly enumerate them without actually proving them at them at this stage variance of two Brownian motion paths is the minimum of time S or T so say I looked I look at my Brownian motion at time s and I look at my Brown in motion at time T and if I try to find covariance of these two terms I will find that the covariance will be the minimum of s and T when here it's clear that s is before T therefore the covariance of these two terms will be s property number 5 says Brownian motion is the Markov process okay so what does it mean Markov process all it means is that past is irrelevant for where I'm going to be in the future ie say we are here so I'm here at this point it's irrelevant what I've done in the past and say irrelevant how I moved in the past for where I'm going to be in the future so past has absolutely no bearing on my future evolution next property broy motion is a martingale okay so this says expected value of Brownian motion which moved by time s so I'm at time T and it moved by Tumnus conditional or all the information conditional filtration ie all the information I have available at time T is just where I am at the moment so via the gain this martingale property can be Illustrated imagine that I'm standing here at time WD I ask myself a question where I'm going to be at the time WT plus s well according to the martingale property of Brownian motion expectation of where I'm going to be at time DT plus s conditional on all the information I have how I moved in the past okay here move up down up and down several times conditional on all the information I have would have done in the past which is denoted by F capital T is nothing else but WT okay so what this means is that I don't expect to have moved at all I expect to stay at the same level ie somewhere here so then negative okay I expect to be at the same point at time T plus s s when I was at time WT finally Brownian motion is continuous everywhere probably on property number seven and differentiable nowhere what this means is that you can probably see it here it continues everywhere because there are no jumps there are no discontinuities in this Brownian motion and you cannot differentiate it anywhere okay so let me erase one more time the chart hold on you may ask well surely this first let's have a look at this bit here surely you can find the derivative of this well my depiction of Brownian motion is a bit inaccurate here because if you actually zoom in to this you would see that it does like this all the time yeah so my Brownian motion goes up and down up and down up and down but actually I can't really find the derivative at any point simply because it's just too rough okay it's got spikes everywhere so for instance if I if I tell you to find a derivative the derivative at this point you will you won't be able to find it because it can be you know is it this is that this is just it just impossible to find derivative and in fact if you zoom if you were to zoom on this Brownian motion path you would see that it just it's constantly moving it's impossible to find a derivative at any point will prove this more formally later on and that's what property number 1/8 is saying growing emotions fractal ie two irregular and rough in structure in the next videos we'll be showing all these properties in more detail actually proving them one by one

Channel: stepbil

Views: 148,742

Rating: 4.9337959 out of 5

Keywords: brownian, motion

Id: 7mmeksMiXp4

Length: 11min 33sec (693 seconds)

Published: Sun Mar 20 2011