(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 X_t = f(t, B_t), and use Ito’s Lemma 8.2 to calculate dX_t. Next, use the chain rule and Ito’s Lemma 8.2 to calculate dY_t, where Y_t = g(t, X_t) = g(t, f(t, B_t)).

B)      Finally, calculate dX_t · dX_t by the box calculus and verify that your expression for dY_t shows that the box calculus formula (8.28) is valid for Y_t.

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