Initial mean and variance derivatives of an Itô diffusion (${{\partial }_t\mathbb{E}{X}_t|}_0=b(x),{{\partial }_t\mathrm{Var}{X}_t|}_0=\sigma (x{)}^2$)

For deterministic $X_0=x$ and bounded continuous drift and volatility, the initial right derivative of the mean is $b(x)$ and that of the variance is $\sigma (x{)}^2$. The stochastic integral has mean zero and second moment ${\int }_0^t\mathbb{E}\sigma (X_s{)}^{2}ds$; the bounded drift contributes only higher-order terms to the variance.

// Target: probability-and-statistics.bigb