Parameter derivative of the exponential Brownian martingale
= Parameter derivative of the exponential Brownian martingale
Every parameter derivative
$$
\frac{\partial^n}{\partial\lambda^n}
\exp(\lambda B_t-\lambda^2t/2)
$$
is 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 $B_t^3-3tB_t$.