 Solution

The gradient bound makes $f$ a Lipschitz function, with $|f(x)|\le |f(0)|+K\Vert x\Vert$, so its value at a Gaussian random variable is integrable and square integrable. The future Brownian increment $X_1-X_t$ is independent of ${\mathcal{F}}_t$ and has distribution $N(0,(1-t)I_d)$. Conditioning on the current location therefore givesThis is the Markov property expressed through the Brownian transition semigroup. In particular $M_0=\mu$ and $M_1=f(X_1)$.