In part (ii), the question involved time s, but in the solution no s is found. My solution contains time s and others (s, s+t-w, s+t). Can we just assume s=0? In part (iii), the solution magically makes s appear again. Please tell me this s have nothing to do with the s in part (ii) because it looked more like the w in part (ii).
We don't need to include s in the calculations because this is a time homogeneous model - the question tells us that mu depends on i but not on the time t. If we had a time inhomogenous model, we would get something like: P i,i+1 (s, s+t) = integral from 0 to t { p i,j-1 (s, s+w) * lambda j-1 (s+w) * p j,j bar (s+w, s+t) } dw As our model is time homogeneous, p i,j (s, s+w) depends only on the length of time between s and s+w. It doesn't matter what time s we start in, we always have the same probability as the force of mortality doesn't change over time. So instead we just say p i,j (w). Similarly, lambda j-1 is constant so we can drop the (s+w), and p jj bar (s+w, s+t) depends only on the length of time (s+t) - (s+w) = (t-w) so we get p j,j bar (t-w). Yes, the s in part (iii) is the same as the w in part (ii), it has nothing to do with the s in part (ii). Hope this helps!
