Let {W}zo be a standard Brownian motion. Let Xt=et,t 20 Y =W-tW,t = [0,1] Zt =...

 

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Let {W}zo be a standard Brownian motion. Let Xt=et,t 20 Y =W₁-tW₁,t = [0,1] Zt = Weds, t≥ 0 VtZt-tWt, t≥ 0 (a) Find P(W₁ + W₁ > 3). (b) Are the increments of the process X, independent? An- swer YES or NO and explain. (c) Calculate Var (X6). (d) Calculate Var (Y3). (e) Calculate Var(Z6). (f) Are Zu and Z, uncorrelated for any u, v> 0? Answer YES or NO and explain. (g) Compute Cov(V5, W5). (h) Give an expression for Corr(W₁, W₁t) for any 0 < t < 1. Let {W}zo be a standard Brownian motion. Let Xt=et,t 20 Y =W₁-tW₁,t = [0,1] Zt = Weds, t≥ 0 VtZt-tWt, t≥ 0 (a) Find P(W₁ + W₁ > 3). (b) Are the increments of the process X, independent? An- swer YES or NO and explain. (c) Calculate Var (X6). (d) Calculate Var (Y3). (e) Calculate Var(Z6). (f) Are Zu and Z, uncorrelated for any u, v> 0? Answer YES or NO and explain. (g) Compute Cov(V5, W5). (h) Give an expression for Corr(W₁, W₁t) for any 0 < t < 1.