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$X$ is a (continuous time) Markov chain with generator matrix $\Lambda$ and finite state space $G$. I know that for $g\colon G \to R$ $$ M_t = g(X_t) - g(X_0) - \int_0^t (\Lambda g)(X_s)\, ds $$ is a martingale. Is it obvious how to extend this to $f\colon[0,T] \times G \to R$ where $f$ is $C^1$ wrt $t$ i.e. to show that $$ M_t = f(t,X_t) - f(0,X_0) - \int_0^t \left(\frac{\partial f}{\partial u}(u, X_u) +(\Lambda f)(u,X_u)\right)\, du $$ is also a martingale? Any help would be appreciated.

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    Maybe we have to assume some boundedness condition on $f$ and its derivative. This result is on Landim's book about particle systems.2012-06-14
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    $f$ is bounded since continuous in $t$ and $G$ is finite. Thanks for the reference, will try to find it.2012-06-14

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