Bounded backward-equation stochastic representation ($u(t,x)={\mathbb{E}}_{x}f(X_t)$)

For a bounded smooth solution of $u_t=Lu$ with initial function $f$, apply the Itô formula to $u(t-s,X_s)$. Its drift vanishes and its boundedness upgrades the localized stochastic integral to a true martingale. Taking endpoint expectations yields $u(t,x)={\mathbb{E}}_{x}f(X_t)$ without a global bound on $u_x$.