Exponential Brownian martingale

For every real $c$,is a martingale. Conditional expectation factors at time $s$ because the Gaussian increment $W_t-W_s$ is independent of the past and has exponential moment $e^{c^2(t-s)/2}$.

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 $B_t^3-3t{B}_t$.