Kolmogorov's backward equation In the book "Stochastic Differential Equations" by Bernt Øksendal, the Kolmogorov's backward equation is stated as following:
Let $X_t$ be an Ito diffusion and $A$ is the generator of $X_t$. Define $u(t,x)=E^x(f(X_t))$ where $E^x$ is the expectation with $X_0=x$. Then we have 
$$
\begin{array}{c}{\frac{\partial u}{\partial t}=A u, \quad t>0, x \in {R}^{n}} \\ {u(0, x)=f(x) ; \quad x \in {R}^{n}}\end{array}.
$$
My question is why this equation is called "Backward"? Sicne now we have the initial condition $u(x,0)=f(x)$. 
Besides, it seems that the definition of Kolmogorov's backward equation is in another form on Wiki: Kolmogorov backward equations (diffusion).
 A: Although it is suggestive to think of forward and backward as the direction of time in the equation, this is slightly misleading. The idea to have in mind should be:


*

*Backward $\leftrightarrow$ initial state $\leftrightarrow$ generator $A$ of the Markov process $\leftrightarrow$ Feynman-Kac.

*Forward $\leftrightarrow$ terminal state $\leftrightarrow$ adjoint $A^*$ of the generator $\leftrightarrow$ Fokker-Planck.
Indeed, you can see that both Oksendal and Wikipedia the operator appearing is $A$. Indeed, if you let $T_{\cdot}$ be the semigroup generated by $A$, you see that:
$$ \mathbb{E}_{0,x}[f(X_t)] = T_t f(x), \qquad \mathbb{E}_{t,x}[f(X_T)] = T_{T-t}f(x)$$
so Wikipedia is just writing the equation for the time-reversed process presented by Oskendal. If you take the probability density of $X_t$ started in $x$ to be $p_t(x,y)$, the above tells you:
$$ \partial_t p_t(x,y) = A_x p_t(x, y), \qquad p_0(x,y)= \delta(x {-} y)$$
Instead, the forward equation tells you that:
$$\partial_t p_t(x,y) = A^*_y p_t(x,y), \qquad p_0(x,y) = \delta(x{-}y).$$
