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If I am not mistaken, the Feynman-Kac formula is related to the Kolmogorov's backward equation, so I would expect it to be available only for Markov processes. Diffusions are usually of Markovian type, in contrast to general Ito processes or more to say, general semimartinagales. Intuitively, the PDE/PIDE/... will describe the dynamics of ...

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