 Parameter derivative of the exponential Brownian martingale

Every parameter derivativeis a martingale. Differentiation passes through conditional expectation because Gaussian exponential moments dominate every derivative locally uniformly in $\lambda$. At $\lambda =0$ these derivatives yield the time-space Hermite martingales, beginning with $B_t$, $B_t^2-t$, and $B_t^3-3t{B}_t$.