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 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.