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Forwards Feynman–Kac formula
Given $t \in (0,T)$, define $\tilde{X}^{(t),x}$ to be the solution of the SDE
\begin{equation*}
d\tilde{X}^{(t),x}_{s} = \mu(t + s,\tilde{X}^{(t),x}_{s}) \, ds + \sigma(t + s, \tilde{X}^{(t),x}_{s}) \ …