A typo in the introduction of Feynman Kac formula

[Wu-Chenyang]

Question❓
I am new to this topic. Forgive me if I am wrong. It seems the second equation in the Feynman Kac formula has a typo. It should probably be
$$u(t, x) = \int_t^T \mathbb{E} \left[ f(X^x_{s-t}, u(s, X^x_{s-t}))ds \right] + \mathbb{E} \left[ u(T, X^x_{T-t}) \right] \tag{2}$$
rather than being
$$u(t, x) = \int_t^T \mathbb{E} \left[ f(X^x_{s-t}, u(s, X^x_{s-t}))ds \right] + \mathbb{E} \left[ u(0, X^x_{T-t}) \right] \tag{2}.$$