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How to prove question 4a

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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) = XitIYitl - 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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Subject: Math
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...

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