SDE of Geometric Brownian motion

Question

I was reading The Binomial Asset Pricing Model by Shreve and having some trouble dealing with SDE. On page 169, it surveys the Geometric Brownian motion and tries to computing the SDE of that process. He says: Define $$f(t,x)=S(0)\exp\lbrace \sigma x+(\mu-\frac{1}{2}\sigma^2)t\rbrace$$ so that the GBM is $$S(t)=f(t,B(t)).$$ According to Ito's formula, $$dS(t) =df(t,B(t)) =f_t dt+f_xdB+\frac{1}{2}f_{xx}dt.$$ But the Ito's formula says for twice differentiable function $F$, we have $$dF(B(u))=F'(B(u))dB(u)+\frac{1}{2}F''(B(u))du.$$ If it is the case $$F(x)=f(t,x),$$ then where does the $f_tdt$ come from?

Answer 1

You added another variable (i.e. explicit time dependence) to the function. For non-stochastic $x(t),$ we have $$ df(x(t)) = f'(x(t))dx(t).$$ When we add explicit time dependence, $f(t,x(t)),$ we need to use the multivariable chain rule, giving $$ df(t,x(t)) = f_tdt + f_x dx(t).$$ Similarly, here, now using the fact that the second argument is a drift-diffusion, $$ df(t,B(t)) = f_t dt + f_x dx(t) + \frac{1}{2}f_{xx} \sigma^2 dt.$$