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How to prove question 4a


4. (5 marks) Ito calculus.
(a) Let X(t) and Y(t) be two stochastic processes, such that
dX = ux(X(t), t )dt + ox (X(t), t )dZx
dy = my (Y (t), t ) dt + oy (Y (t), t)dZy
with Zx (t), Zy(t) being Brownian motions. Let X(t;) = X; and Y(t;) = Y. Show that
(Xit1 - Xi)(Yitl - Yi) = XitIYitI - XiYi - Xi( Yit1 - Yi) - Yi(Xit1 - Xi) .
Now, using the definition of the Ito integral which is the limit of a discrete sum, show that
X(s)dY (s) = [Xy]b - Y(s)dX(s) - /dX(s)dY(s).

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