= Solution
The multidimensional <Itô formula> includes a mixed second derivative multiplied by the <quadratic covariation>, with no extra factor $1/2$. Here
$$
d[\sigma]_t=B(\sigma_t)^2dt,\qquad d[X]_t=\sigma_t^2dt,\qquad d[\sigma,X]_t=\rho\sigma_tB(\sigma_t)dt.
$$
Consequently the <Itô formula> for $M_t=U(t,\sigma_t,X_t)$ has drift equal to the left side of the stated backward <partial differential equation>. That drift vanishes, leaving
$$
\boxed{dM_t=B(\sigma_t)U_\sigma(t,\sigma_t,X_t)dW_t^\sigma
+\sigma_tU_X(t,\sigma_t,X_t)dW_t^X.}
$$
A <stochastic integral> against <Brownian motion> with locally square-integrable predictable integrand is a continuous <local martingale>. The smoothness of $U$ and localization of the diffusion and its coefficients give this integrability on the model's lifetime. Thus $M$ is a <local martingale>, as required. The <partial differential equation> cancellation alone does not establish a true <martingale> or justify replacing $U$ by a terminal-payoff expectation without an additional integrability argument.
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