= Solution
Let $u(t,x)=P_{1-t}f(x)$ for $t<1$. Differentiating the <Brownian transition semigroup> in the spatial variables is justified by the bounded gradient and gives
$$
\partial_i u(t,x)=P_{1-t}(\partial_i f)(x).
$$
The <heat equation> for this <semigroup> is the backward equation $\partial_tu+\tfrac12\Delta u=0$. Applying the multidimensional <Itô formula> yields
$$
du(t,X_t)=\left(\partial_tu+\tfrac12\Delta u\right)(t,X_t)\,dt
+\sum_{i=1}^d\partial_i u(t,X_t)\,dX_t^i.
$$
The finite-variation term vanishes. Thus
$$
\boxed{dM_t=\sum_{i=1}^d P_{1-t}(\partial_i f)(X_t)\,dX_t^i.}
$$
Initially this calculation holds on $[0,1-\varepsilon]$. The vector of <stochastic integral> coefficients has norm at most $K$, so the <Itô isometry> lets the integrals extend to time one. Also $P_{1-t}f(X_t)\to f(X_1)$ in $L^2$ by the <Lipschitz function> bound and continuity of <Brownian motion>. The same representation therefore holds on the entire interval.
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