Basic Properties of Standard Brownian Motion - Essay Example

Basic properties of standard Brownian motion Property One and Two The first basic property of standard Browinian motion, de d as W0 indexed with time, is that W at time 0 equals 0. This means that Brownian motion at time = 0 takes the value of zero. Brownian motion can take a variety of paths. However, one thing that all realisations of Brownian motion have is that they take a value of 0 at time = 0. So, it can be denoted by W at time 0=0. Clearly, this property is pretty much self explanatory. The second property is that Brownian motion is normally distributed with mean 0 and variance t and Brownian motion increment Wt - Ws is also normally distributed with mean 0 and variance (t-s). What does it mean to say that the Brownian motion is normally distributed with mean 0 and variance t? At a point in time, say time = 0, knowing what Brownian motion is or will be at time, t=2. The Brownian motion can take on a number of single realisations, that is, it can take a range of values. What is known is the expected value to the distribution of Brownian motion at time =2. Therefore, the centre of the distribution is known, i.e. what the expected value of the distribution is and this will be the expected value of W2= 0. It will always be zero, regardless of what point in time we view the Brownian motion. The expectation of Brownian motion at all points on a plain at any time is 0 as per property one. Not only will the expected value at any time be 0, but also normally distributed. The peak of the normal distribution is centred at 0, meaning that the Brownian motion will be distributed as a normal variable with expected value 0 and variance t. Property Three Property three relates to the concept of property number two, i.e. the Brownian motion increment, which is the difference between the two Brownian motions (Wt - Ws). Therefore, the difference between the two Brownian motions is also normally distributed and the variance of the Brownian motion increments (Wt - Ws) is (t-s), where t stands for time and s stands for a point in time which differs from t. (t - s) is the difference in two time periods between measurements of our Brownian motion. Consequently, looking at the Brownian motion at two different points in time, the expected increment , the expectation of the difference of these two Brownian motions ( E [Wt - Ws])=0 and the variance of this difference ( Var [Wt-Ws]) = t-s. It emerges that the variance is proportional to time. Other properties of Brownian motion state that the process Wt has stationary and independent increments. What does it mean to say that the Brownian motion has stationary increments? Looking at an example of a Brownian motion at time = 0 (W0) and the same Brownian motion at time = 1 (W1) and then looking at a graph of our Brownian motion , it moves the Brownian motion increment further in time by a constant amount (a). This will be W0+a and W1+a and what this means is that the distribution of the increment (the difference between W1 and W0) will be exactly the same as the distribution of this increment (the difference between W0+a and W1+a) i.e. they will both be normal with expected value of 0 and variance of (t1-t0) N~ (0, t1- t0). As such, regardless of where you look at the Brownian motion on a graph, it will always have the same properties. The incremental expectation will always be 0 and the variance will be the difference in time. To sum up, where you view the Brownian motion does not matter, it will always have the same property in a probabilistic sense. It will always have an expectation of 0 and variance will be proportional to time. Another property is that increments are independent, that is, if W1 is less than W0 and W1+a less than W0+a. Independence means that the motion moves in a time interval (between 1 and 0) and will have absolutely no bearing on how it will move in the next time interval (1+a and 0+a). Property Four Looking at some other properties of Brownian motion, the covariance of two Brownian motion paths is the minimum of time s or t. (cov{W(t), W(s)} = minimum{s, t}. Looking at a Brownian motion at time s and looking at a Brownian motion at time t, and if one tries to find covariance of these two terms (Ws and Wt), you will find that the covariance will be the minimum of s and t. If time s comes before time t, then the minimum of the two terms is s. Property Five Property number five states that Brownian motion is a Markov process. This means that the past is irrelevant for where the process will be in the future. It is irrelevant where the process has been in the past for deciding on where it will be in the future. The past has absolutely no bearing on the future evolution of the process. Property Six Property six states that Brownian motion is a martingale.The expected value of Brownian motion, which moved by time s ( it is at time t and moved by time s) is conditional on all the information at time t and is where the process is at the moment, at Wt. This martingale property can be illustrated. Imagine that the process is at time Wt .Where is the process going to be at time Wt+s?According to the martingale property of Brownian motion, the expectation of where the process is going to be time t + s is conditional on all the information known of how the process moved in the past is equal to Wt . What this means is that the process remains stationary. It stays at the same level i.e. at the same point at time t + s, as when it was at time Wt. Property Seven Brownian motion is continuous everywhere and differential nowhere. What do we mean by saying it is continuous everywhere? There are no jumps there are no discontinuities in Brownian motion and you cannot differentiate it anywhere. One cannot find a derivative of the Brownian motion as it is too volatile, it has spikes everywhere. It is impossible to find the derivative and in fact, if one zooms in on a Brownian motion path you would see that it is constantly moving. Property Eight Property number eight states that Brownian motion is fractal, that is, it is irregular and rough in structure. Read More