calculate dXt · dXt by the box calculus and verify that your expression for dYt shows that the box calculus formula (8.28) is valid for Yt.

(A Box Calculus Verification). The purpose of this exercise is to generalize and unify the calculations we made for functions of Brownian motion with drift and geometric Brownian motion. It provides a proof of the validity of the box calculus for processes that are functions of Brownian motion and time.

A)     Let Xt = f(t, Bt), and use Ito’s Lemma 8.2 to calculate dXt. Next, use the chain rule and Ito’s Lemma 8.2 to calculate dYt, where Yt = g(t, Xt) = g(t, f(t, Bt)).

B)      Finally, calculate dXt · dXt by the box calculus and verify that your expression for dYt shows that the box calculus formula (8.28) is valid for Yt.

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