I am reading Stochastic Processes and Applications by Pavliotis. At some point he defines Kolmogorov backward equation as the differential equations

$$\frac{\partial u}{\partial t} = \mathcal{L}u$$ $$u(x, 0) = f(x)$$ where $u(x, t) = E[f(X_t)|X_0 = x]$ and $\mathcal{L}$ is the generator of $X_t$.

Later he defines Kolmogorov backwards equation as an actual backward (in time) equation for $u(x,t) = E[f(X_T) | X_t = x]$.

Kolmogorov's forward equations are defined in terms of the adjoint operator acting on probability measures, always going forward in time.

Which one of the above is the actual Kolmogorov backwards equation and why is it called backward/forward?


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