Exercise 3.6 Let 8(t), I 3 0 be a geometric Brownian motion process with drift parameter ,1 and volatility parameter or. Assuming that 8(0) = s, nd Var(8(t)). Hint: Use the identity Var(X) = E[X2] (EIXD2 Exercise 3.7 Let {X (I), t 3 0} be a Brownian motion process with drift parameter p, and variance parameter 02. Assume that X (0) = 0, and let Ty be the rst time that the process is equal to y. For y > 0, show that , 1, 1f u 3 0 P T : Let M = maxudoo X (t) be the maximal value ever attained by the process, and conclude from the preceding that, when p, < 0, M has an exponential distribution with rate 2,u,/02. Exercise 3.8 Let 8(1)), 1) Z 0 be a geometric Brownian motion process with drift parameter ,u and volatility parameter or, having 8(0) = 3. Find P (maXOEugt 8(1)) 3: y). Exercise 3.9 Find P(max05,,51 8(1)) <: 1.2 8(0)) when 8(1)), 1) 3 0, is geometric Brownian motion with drift .1 and volatility .3