3.6 Exercises Exercise 3.1 If X (I), 1' 3 0 is a Brownian motion process with drift parameter ,u. and variance parameter a2 for which X (0) = 0, show that X (0,: Z 0 is a Brownian motion process with drift parameter ,u. and variance parameter 02. Exercise 3.2 Let X (t), t 2 0 be a Brownian motion process with drift parameter p. = 3 and variance parameter 0'2 = 9. If X (0) = 10, nd (a) EIX (2)]; (b) Var(X (2)); (c) MK (2) > 20); (d) P(X(.5) 2. 10). Exercise 3.3 Let A = 0.1 in the approximation model to the Brown- ian motion process of the preceding problem. For this approximation model, nd (a) E [X (1)]; (b) Var(X(1)); (c) P(X(.5) > 10). Exercise 3.4 Let SU) , I Z 0 be a geometric Brownian motion process with drift parameter ,u. = 0.1 and volatility parameter or = 0.2. Find (a) P(S(1) 2* 5(0)); (b) P(S(2) > 3(1) > 5(0)); (c) P(S(3) <: 5(1) > 3(0)). Exercise 3.5 Repeat Exercise 3.4 when the volatility parameter is 0.4