Question: 3. Let (2, F, P) be a probability space and let {W(),t a. Show that X(t) W2(t) -t is a martingale. (Hint: show that E[X(t)F(s)

3. Let (2, F, P) be a probability space and let

3. Let (2, F, P) be a probability space and let {W(),t a. Show that X(t) W2(t) -t is a martingale. (Hint: show that E[X(t)F(s) -X(s)) 0 be Brownian motion with respect to the filtration Ft, t 20 b. For R, show that the following hyperbolic processes: X(t)cosh(AW(t) is a martingale. (Hint: Use the identity cosh(z)t-) c. Show that d. Show that X(t) W(t) 3t is a martingale. f. Show that Corr W-(t), W-(s 3. Let (2, F, P) be a probability space and let {W(),t a. Show that X(t) W2(t) -t is a martingale. (Hint: show that E[X(t)F(s) -X(s)) 0 be Brownian motion with respect to the filtration Ft, t 20 b. For R, show that the following hyperbolic processes: X(t)cosh(AW(t) is a martingale. (Hint: Use the identity cosh(z)t-) c. Show that d. Show that X(t) W(t) 3t is a martingale. f. Show that Corr W-(t), W-(s

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