3. (12 marks) (Ito’s Calculus) Suppose dZ is an increment of a standard Brownian motion. An Ito’sprocess satisfies the following stochastic differential equationdX = a ( X, t) dt + b (X, t) dz.Ito’s integral f, b(Z (t) , t) dZ (t) is the mean square limit of the Ito’s sum, i.e.N-1limN-++00b(Z (tn) , tn) (Z (tin+1) – Z (tn))n=0where tn is defined in (1).(a) Write down the Ito’s sum for f, Z?dZ (t) where Z? is the square of Z (t).(b) What is the expected value of this Ito’s sum?(c) LetY (t) = e* +3Z(t)show thatdY (t) =2t +OINY (t) dt + 3Y (t) dZ (t) .
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