Initial-value derivative of a linear Gaussian diffusion (${\partial }_x\mathbb{E}f(X(t,x))=a(t)\mathbb{E}{f}^{\prime }(X(t,x))$)

For $dX=\mu (t)Xdt+\sigma (t)dB$ with deterministic coefficients, $X(t,x)=a(t)[x+{\int }_0^{t}a(s{)}^{-1}\sigma (s)d{B}_s]$, where $a(t)=\exp ({\int }_0^t\mu )$. The random part does not depend on the initial value. For $f\in C_b^1$, the mean value theorem bounds the initial-value difference quotient by $a(t)\Vert f^{\prime }{\Vert }_{\infty }$, and dominated convergence theorem gives the displayed derivative. The transition law is normal with mean $a(t)x$ and variance ${\int }_0^t\exp (2{\int }_s^t\mu )\sigma (s{)}^{2}ds$.