please solve complete in one hour

B5 Assume that (B7, t > 0) is a standard Brownian motion. Consider the two Geometric Brownian motions: )t+oB Xi = elu- -*t Y; = e(123)t+02B! 2- so that X4 has drift parameter jy and variance parameter 01, whereas the corresponding parameters for Y4 are p2 and 02. (a) Let Rt = X+ +Y4. Show that when Mi = M2 = u and 01 = 02 = 0 > 0, then Rt is also a Geometric Brownian motion. Give the initial condition, the drift parameter and the variance parameter for Rt. (b) Let S = X+ Yt. Show that St is a Geometric Brownian motion under any choice of ui, H2 and any choice of 01 > 0,02 > 0. Give the initial condition, the drift parameter and the variance parameter for St. B6 Assume that (Bt, t > 0) is a standard Brownian motion. = (a) Let Xt = V1B1. Is Xt a Brownian motion? Explain briefly your answer. (b) Let Y{ = (1+t)B 4 - tB1. Compute the expectation and covariance of Y4. Is Y; a Brownian motion? Explain your reasoning. t+1 B5 Assume that (B7, t > 0) is a standard Brownian motion. Consider the two Geometric Brownian motions: )t+oB Xi = elu- -*t Y; = e(123)t+02B! 2- so that X4 has drift parameter jy and variance parameter 01, whereas the corresponding parameters for Y4 are p2 and 02. (a) Let Rt = X+ +Y4. Show that when Mi = M2 = u and 01 = 02 = 0 > 0, then Rt is also a Geometric Brownian motion. Give the initial condition, the drift parameter and the variance parameter for Rt. (b) Let S = X+ Yt. Show that St is a Geometric Brownian motion under any choice of ui, H2 and any choice of 01 > 0,02 > 0. Give the initial condition, the drift parameter and the variance parameter for St. B6 Assume that (Bt, t > 0) is a standard Brownian motion. = (a) Let Xt = V1B1. Is Xt a Brownian motion? Explain briefly your answer. (b) Let Y{ = (1+t)B 4 - tB1. Compute the expectation and covariance of Y4. Is Y; a Brownian motion? Explain your reasoning. t+1