Let (W)to be a Brownian Motion. (a) Let 1 if Wt > 0, Xt=sgn(W)= 0 t if W₁ = 0, - 1 if Wt < 0. Check that (X+) is an Itô-integrable process. (b) Let Y₁ = [X, W. Find E[Y₁] and Var(Y;). 0 (c) Check that (W2) is an Itô-integrable process. (d) Consider the function f : [0,T] → R defined by f(t) = ta, where a € R. For which f(s) dW, well-defined? When it is, find E[Z₁] values of a is the process Z₁ = and Var(Z). Bonus question: In part (c), consider a partition 0 = to < ··· < tn = T. Prove that n-1 the simple process W221 +) (+) converges to W2 in the L²-sense. i=0 1