= Dynkin's formula
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{title2=$f(X_t)-f(X_0)-\int_0^tLf(X_s)ds\text{ is a local martingale}$}
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= Dynkin formula
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{synonym}
For a <Markov process> with <infinitesimal generator> $L$ and a suitable <test function> $f$, the displayed process is a <local martingale>. With integrability it is a true <martingale> and yields the corresponding <expected value>[expectation] identity at admissible <stopping time>[stopping times]. For bounded $f,Lf$ in the semigroup <operator domain>[generator domain], $P_tf-f=\int_0^tP_sLf\,ds$. The <Markov property> applies this identity conditionally over each <interval>, proving the <martingale> increments have zero <conditional expectation>. Localization gives the local version. For a continuous <diffusion process>[diffusion], these identities are the <martingale problem> for $L$.
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