Hello,I have a small problem with how Ito's Lemma is applied to obtain an explicit expression for the price (S) of a security from the Brownian motion, here is the standard derivation present in many texts (including Hull):Brownian Motion Ito's LemmaApplying Ito's LemmaThe part I don't understand, is how the the term (dG/dt) is set to zero? S is by definition a function of time, thereby any function of S is also a function of time (in this case G)

Careful ...G should be seen as a function of two variables, x and t : G(x, t).Here G(x,t) = ln(x). Then dG/dt = 0.

This is why people use different symbols for partial and total derivatives...partial of G wrt t is zero since by definition S is kept constant when taking the partial

That does make sense .... thanks for your replies

Here G(x,t) = ln(x). Then dG/dt = 0. But we know that x is a function of t, x = f(t). So if we simply substitute G(x,t) = ln(x) = ln(f(t)) = G(f(t), t), then G becomes a function of t. I am having trouble convincing myself that the dependence on t somehow vanishes when you define an intermediate variable x, because I can always substitute x with a function of t. I must be wrong since every single textbook says dG/dt = 0. Any help on the intuition is appreciated!

is this the same judge Ito from the OJ trial?

QuoteOriginally posted by: ezbentley I must be wrong since every single textbook says dG/dt = 0. Any help on the intuition is appreciated!S is not a function of time; S is a stochastic process -that is, a collection of random variables {S_t}_t. For each t>=0, you have S_t. The values S_t may take depend on the possible scenarios in your particular model.

QuoteOriginally posted by: manolomQuoteOriginally posted by: ezbentley I must be wrong since every single textbook says dG/dt = 0. Any help on the intuition is appreciated!S is not a function of time; S is a stochastic process -that is, a collection of random variables {S_t}_t. For each t>=0, you have S_t. The values S_t may take depend on the possible scenarios in your particular model.Thanks. I was thinking along more or less the same line. S can take on different values at different time, but it is not a "function" of time in the traditional sense. You cannot plug in a value of time and get a deterministic value for S.