Solution
= Solution
Apply the <Itô product rule> to the deterministic discount factor and $u(X_t)$:
$$
d(e^{-\lambda t}u(X_t))=e^{-\lambda t}(\mathcal Lu-\lambda u)(X_t)\,dt+e^{-\lambda t}\nabla u(X_t)^{\mathsf T}\sigma(X_t)\,dW_t.
$$
The prescribed differential equation makes the drift vanish. Therefore
$$
\boxed{M_t=e^{-\lambda t}u(X_t)\text{ is a local martingale}.}
$$
This is the discounted generator-eigenfunction martingale underlying the <Feynman-Kac formula>.