Suppose (W; t > 0) is a standard Brownian motion under P (with its expectation E)...

 

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Suppose (W₁; t > 0) is a standard Brownian motion under P (with its expectation E) and (FW; t > 0) denotes the filtration generated by W. (a) (2 marks) Compute the expectation of the absolute value E[W] for t > 0. (b) (3 marks) Show that Xt= W² - t, t≥0, is a martingale with respect to (FW;t > 0). For this exercise, do NOT use Itô's rule. Directly show the properties given in L8.11 (see also L8.12). (c) (3 marks) The Itô isometry says, for any process (Yt;t0) adapted to (FW;t > 0), B. [(* Yaw,)*] - R[/" Vedi]. E Y+dWt = E By using the Itô isometry and the results given in L8.40, show that Also compute E Widt = E[WY "taw.]. tdWt Suppose (W₁; t > 0) is a standard Brownian motion under P (with its expectation E) and (FW; t > 0) denotes the filtration generated by W. (a) (2 marks) Compute the expectation of the absolute value E[W] for t > 0. (b) (3 marks) Show that Xt= W² - t, t≥0, is a martingale with respect to (FW;t > 0). For this exercise, do NOT use Itô's rule. Directly show the properties given in L8.11 (see also L8.12). (c) (3 marks) The Itô isometry says, for any process (Yt;t0) adapted to (FW;t > 0), B. [(* Yaw,)*] - R[/" Vedi]. E Y+dWt = E By using the Itô isometry and the results given in L8.40, show that Also compute E Widt = E[WY "taw.]. tdWt