GBM Essay Example | Topics and Well Written Essays - 1000 words

GBM - Essay ExampleWhat is known is the expected value to the diffusion of Brownian motion at snip =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 horizon the Brownian motion. The expectation of Brownian motion at all points on a plain at any time is 0 as per proportion one. Not exclusively will the expected value at any time be 0, but also normally distributed. The peak of the normal distribution is centred at 0, convey that the Brownian motion will be distributed as a normal variable with expected value 0 and variance t.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 inc rements (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 unlike 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 get along in time by a constant amount (a). This will be W0+a and W1+a and what this means is that