> [!tldr] > If an underlying $(X_{t})$ evolves according to $dX_{t}=\mu ~dt + \sigma~dW_{t},$where $(W_{t})$ is a standard [[2 - Snippets/Brownian Motion]], and $\mu, \sigma$ can be functions dependent on $X, t$, then a function $f(X_{t}, t)$ has the dynamics $df = \left( \frac{ \partial f }{ \partial t } +\mu \frac{ \partial f }{ \partial X_{t} } +\frac{\sigma^{2}}{2} \frac{ \partial^{2} f }{ \partial X_{t}^{2} } \right)dt + \sigma\frac{ \partial f }{ \partial X_{t} }dW_{t}.$