= Initial mean and variance derivatives of an Itô diffusion
{title2=$\left.\partial_t\mathbb EX_t\right|_0=b(x),\quad\left.\partial_t\operatorname{Var}X_t\right|_0=\sigma(x)^2$}
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 $\int_0^t\mathbb E\sigma(X_s)^2ds$; the bounded drift contributes only higher-order terms to the variance.
// Target: probability-and-statistics.bigb
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