Ito's chain rule and multidimensional derivation

Question

In the book "Quant Job Interview: Questions and Answers" by M. Joshi in the derivation of the final Black-Scholes formula he makes usage of Ito's chain rule. To get specific, he goes from
$$ dC (t, S_t) = \left(\frac{\partial C}{\partial t} + \frac{\partial C}{\partial S_t} r S_t + \frac{1}{2} \frac{\partial^2 C}{\partial S_t^2} \sigma^2 S_t^2\right)dt + \sigma S_t \frac{\partial C}{\partial S_t} dW_t$$ to $$ d\left(\frac{C (t, S_t)}{B_t} \right) = \frac{1}{B_t}\left(\frac{\partial C}{\partial t} + \frac{\partial C}{\partial S_t} r S_t + \frac{1}{2} \frac{\partial^2 C}{\partial S_t^2} \sigma^2 S_t^2 - rC\right)dt + \sigma \frac{S_t}{B_t} \frac{\partial C}{\partial S_t} dW_t.$$ In his book he shortly claims he has achieved that by simply applying Ito's chain rule. I'm familiar with the standard case of Ito's Lemma, but I don't quite get which connection there is in this case. Thanks in advance.

Answer 1

It is most likely what is called Ito's product rule or Leibniz rule; given two (one dimensional) Ito processes $$ dX_t=\mu_{1,\,t}dt+\sigma_{1,\,t}dW_t $$ and $$ dY_t=\mu_{1,\,t}dt+\sigma_{1,\,t}dW_t, $$ the differential of their product is $$ d(XY)=XdY+YdX+dXdY. $$

In your case, you can do the computations considering that, by differencing $f(B_t)=\frac{1}{B_t}=e^{-rt}$ $$ d\frac{1}{B_t}=f_tdt=-re^{-rt}dt=-r\frac{1}{B_t}dt. $$