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3. Let {Bt, t > 0} denote a standard Brownian motion. (a) Find E[X Y], where [5 marks] t Xt = e 2Bs d Bs,

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3. Let {Bt, t > 0} denote a standard Brownian motion. (a) Find E[X Y], where [5 marks] t Xt = e 2Bs d Bs, Yt= Bs dB s. 0 (b) Show that Xt = BA - 6tB2 + 3t2 is a martingale. [5 marks] (c) Let T = inf{t > 0 | B? + 2t =6}. Determine E[r] and E[72]. [5 marks]

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