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For the Kolmogorov backward and forward (aka Fokker-Planck) equations to hold, and also for the Feynman-Kac formula, is it necessary for the terms in the stochastic differential equation $$ dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t $$ to be time-independent $$ dX_t = \mu(X_t)dt + \sigma(X_t)dW_t $$ or not? My impression is that the requirement is markovity, and thus one often finds the theorems in terms of Itô's diffusions, which are always Markov processes.

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