Write 4 pages thesis on the topic gbm. 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 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).