Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 4 / b / Solution

Differentiate the conditional identity from part a. To justify doing so, fix a compact parameter interval $|\lambda |\le L$. Every $n$th derivative of $M_{\lambda }(t)$ is a polynomial in $B_t,t,$ and $\lambda$ times $M_{\lambda }(t)$, and its absolute value is bounded byThis bound is integrable because a Gaussian random variable has every polynomially weighted exponential moment. Dominated differentiation of conditional expectation therefore givesThus every parameter derivative of the exponential Brownian martingale is itself a martingale.