Ito's lemma

When X follows $dX_t = \mu_t\,dt + \sigma_t\,dW_t$, the change in f is the usual terms in time and X plus the term $\tfrac{1}{2}\sigma_t^2\,\partial^2 f/\partial x^2$ multiplied by $dt$. This term arises because the square of a Brownian increment is replaced by $dt$.

For example, if $f = \ln S$ and $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$, the drift of $\ln S$ is $\mu - \sigma^2/2$. Consider a hypothetical example in which $\mu = 5\%$ and $\sigma = 20\%$. The drift is $0.05 - 0.04 \div 2 = 0.03$.

If the function is linear, the extra term is 0 and the result is the same as the ordinary chain rule. The derivation and worked examples are on the related article page.