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Consider the function f(x) of x for which the partial derivatives de le, and defined and continuous

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Consider the function f(x) of x for which the partial derivatives de le, and defined and continuous. one-dimensional Itô rule df (X.) = (X.)dx, + (W) (dx,)?, where {Xt}te(0,1) is an Itô process. Here, the notation (Xt) and (X4) means that the partial derivatives are evaluated at Xt. We consider the special case Xt = W4 under which the above Itô rule becomes af . 1a2f df (W) -(W+)dW+ + (W+)dt. ?? 2 ax2 Integrating the above from t= 0 to t=T gives af (W+) dt. (1) ax ?r2 In this question, you are asked to prove (1). To this end, for fixed T > 0 and fixed positive integer n, with An T/n, let IIn {ti}_o, where t; iAn, be a partition of the interval [0,T]. To simplify the notation, we set AWt; = Wtit1 – Wti, i = 0,..., n - 1. Prove (1) for the case f(x) = 2x2. (Wr) – 5 (W) = 6" (w.) aw. + = a. Hint: For fixed n, use n-1 f (WT) – f (Wo) = (f (Wi+1) – f (W;)), i=0 and then use Taylor's Theorem for each term f (Wi+1) – f(Wi). Finally, find the limit of the sum as n + O. b. Mathematically prove that (1) still holds for an (sufficiently smooth) arbitrary function f(x). Hint: Use the same approach as in part (a). A key difference between part (a) and part (b) is that in part (a), since f(x) = 2x2, it follows that one 1; however, in part (b), this is not necessarily the case.