The following screenshot shows part of the provided lecture notes for a module on actuarial mathematics:
My question is as follows: is this equation equivalent to simply writing $$ \bar{A}_x^{ik} = \int_0^\infty e^{-\delta t} {}_t p_x^{ik} \hspace{1mm} dt $$
Why bother to add the additional consideration of first moving into an unrelated state $j$ at time $t$, rather than simply considering the transition directly into state $k$ from the initial state $i$?
Either the expression in the lecture notes is unnecessarily convoluted, or my expression is incorrect. If the latter is the case, what makes my expression wrong?
If I understand the notation, your simplification is not correct. $_tp_x^{ik}$ is the probability that a life aged $x$ in state $i$ at time $0$ is in state $k$ at time $t$. But the instrument pays at the moment of transition to state $k$. Your integral would have it paying at all times when the life is in state $k$.