= Initial-value derivative of a linear Gaussian diffusion
{title2=$\partial_x\mathbb E f(X(t,x))=a(t)\mathbb E f'(X(t,x))$}
For $dX=\mu(t)Xdt+\sigma(t)dB$ with deterministic coefficients, $X(t,x)=a(t)[x+\int_0^ta(s)^{-1}\sigma(s)dB_s]$, 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 $a(t)\|f'\|_\infty$, and <dominated convergence theorem> gives the displayed <derivative>. The transition law is normal with mean $a(t)x$ and variance $\int_0^t\exp(2\int_s^t\mu)\sigma(s)^2ds$.
Back to article page
