= Solution
Define the <diffusion generator> $L=b\partial_x+\sigma^2\partial_{xx}/2$. Fix a horizon $t$ and start the strong solution from the deterministic state $x$. The <Itô formula> applied to the time-reversed test function gives
$$
d\,u(t-s,X_s)=\bigl[-u_t+Lu\bigr](t-s,X_s)\,ds+\sigma(X_s)u_x(t-s,X_s)\,dW_s.
$$
The drift vanishes by the <Kolmogorov backward equation>. This is initially a <local martingale>; localization on compact state/time sets justifies the <stochastic integral> without a global derivative bound. Since $u$ itself is bounded, this <local martingale> is a true <martingale> on $[0,t]$. Its two endpoint expectations give
$$
\boxed{u(t,x)=\mathbb E_x[f(X_t)].}
$$
This is the <bounded backward-equation stochastic representation>. Starting the strong solution at deterministic $x$ is the precise meaning of the conditional notation at $X_0=x$. Also $f=u(0,\cdot)$ is bounded, even though boundedness was not separately imposed on $f$.
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