Lesson 20 proof of the alternative formula for EPV of continuous annuity
In lesson 20, we have direct way for EPV of continuous whole life annuity:
a_x=∫ a_t*t_p_x*μ_x+t dt, integral from 0 to infinite
The textbook says to plug a_t=(1v^t)/ δ into the equation above and apply integration by parts, hence the new equation:
a_x=∫ (1v^t)/δ*fx(t) dt since t_p_x*μ_x+t = fx(t)
= 1/δ * [∫fx(t) dt  ∫ v^t*fx(t) dt]
v^t is the discount factor, which equals to e^(δt)，the first integral ∫fx(t) dt equals to 1 since I could split the integral into [0,t] and [t,∞], which gives me Fx(t) and Sx(t) that sum up to 1, so the equation becomes:
a_x=1/δ * [1 ∫ v^t*fx(t) dt]
now integration by parts,
u = e^(δt) v= 1 as shown above
du=δ*e^(δt)dt dv=fx(t) dt
so ∫ v^t*fx(t) dt = e^(δt)  ∫δ*e^(δt)dt
I realize that the solution is not the same as the alternative formula:
a_x = ∫ v^t*t_p_x dt
Any help on this one?
