(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.