I Stochastic calculus: Ito's lemma and differentials

[cppIStough]

TL;DR Summary

Ito's lemma and differentials: what's the difference?

Ito's formula states for some stochastic function F(S,t) where S evolves as dS = f(W,t)dt + g(W,t)dW and W is brownian motion:
$$dF = \partial_t F dt + \partial_xF dx + 1/2 \partial_{xx}F dt$$
So why question is, what does dF really mean here? I see in brownian motion we take $$dS = r dt + \sigma dW$$, but simulating this directly via monte carlo gives problems. Evidently the correct approach is to let F = log(S) and apply Ito's lemma. But why can't we just use $$dS = r dt + \sigma dW$$? I mean, it's an equation given, so why doesn't this work with monte carlo?

 

[cppIStough]

Did i put this in the wrong section?

 

[WWGD]

Ito's Lemma tells you that the Ito Differential of an Ito process is itself an Ito process and describes the form of the latter. The Differential describes an infinitesimal change in the Stochastic process in question.

 

[jackjack2025]

I think some of your formulae are maybe not what you intended. You refer to log(S) as though you are doing some Black Scholes type calculation.

W is the Brownian motion in your examples. Based on what you posted, this
dS_t = \mu dt + \sigma dW_t
means
S_t - S_0 = \mu * ( t - 0 ) + \sigma * ( W_t - W_0)

I put in the 0 and the W_0 just for explanation, but of course both are zero and can disappear.

If you meant the Black-Scholes thing, you include S in both the drift and volatility terms, and of course Monte Carlo works, but I don't know which part you are not clear on