Problem 3. Let W := (W+:t > 0) be a standard Brownian motion, i.e, Brownian motion with o2 = 1. - (a) Define X4 = e-t We

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Problem 3 Let W W T 0 Be A Standard Brownian Motion I E Brownian Motion With O2 1 A Define X4 E T We 1 (141.27 KiB) Viewed 66 times

Problem 3. Let W := (W+:t > 0) be a standard Brownian motion, i.e, Brownian motion with o2 = 1. - (a) Define X4 = e-t Wet. Find the mean and the autocorrelation function of X. = (b) Find the conditional probability P({W1 + W2 > 2}|{W1 = 1}). = (c) As in the class, we define the random variable via the integral (m.s): Z := Š W dt. Find Ê[212] (Defined in the class := E[W2] + Cov(W2, Z)Cov(ZZ)-1(Z – E[Z])) (d) Let (Y+ : t > 0) defined by Y1 = W3. Determine whether or not Yis a martin- gale, that is, whether or not E[Y2|Y1] = Y1.