Thiele's Differential Equation - General Reasoning

I'm stuggling to follow the Core Reading general reasoning approach to Thiele's differential equation (Chapter 5, p38). In particular, near the bottom of the page, it says
"Then it is easily seen that, over a short time period dt, the above equation becomes
(t+dt)_[/SIZE]V_x - t_V_x = {t_V_x*delta*dt + P_x*dt - (1-t_V_x)*mu_x+t*dt} + o(dt)"

Can anyone explain how this is derived from the difference equation at the top of the page:
t-1_V_x - t_V_x = 1/(P_x+t) * {i_t*V_x + (1+i)*P_x - q_x+t*(1-t_V_x)}

It doesn't seem "easily seen" to me!

Thanks

 

Funny you should mention that bit, I was struggling with it today. The best I can come up with is
1) it can't follow from the difference equation, they're just analogous.
2) I can't see any way of seeing either directly, I have to start with the "general reasoning" approach described on p39-40 of the notes, which is analogous to 5.14 in the discrete case.

Maybe they have some idea of "setting 1 equal to dt" in the difference equation - for example,
p_x =1_p_x becomes dt_p_x becomes 1- mu_x*dt +o(dt).
(1+i)=exp(delta*1) becomes exp(delta*dt) becomes (1+delta*dt) +o(dt)

Anyone feeling more enlightened?
L

 

I think the corrollary is a bit easier to see if you rearrange the discrete recursion at the top of the page. It may be rearranged with a little algebra to be:

(t+1)_V - t_V = i*t_V + P*(1+i) - q_(x+t)*(1 - (t+1)_V)

Then term by term, we can look at the corrollaries. In general we are replacing the unit of time changed from a discrete unit of 1 to a 'very short' period of time dt. We also add a bar over the top to denote continuous terms, change interest i to a rate delta and multiply rates by time periods dt (ie rate of preium payment or rate of death).

(t+1)_V and t_V -----> (t+dt)_Vbar and t_Vbar
i*t_V --------> delta * t_Vbar * dt
P*(1+i) ------> Pbar * dt (we assume the interest on the premium is included in the o(dt) term)
q_x+t -------->mu_(x+t) * dt
(1 - (t+1)_V) ---------->(1 - (t+dt)_Vbar)

Note the last Vbar is (t+dt) not t. In the notes they have t. I think this is a typo. By justification I would refer you to the third equation of page 40, which is exactly the same, except everything is divided by h (where h represents dt), the q/h has not yet been transformed to mu by taking limits and they've added in the S term.

Technically I think they shouldn't really be writing mu yet either. It should only transform to mu AFTER the limit is taken. Technically I don't reckon they should use the bar notiation until they've taken a limit either.

But its only an illustrative equation, I don't think it makes sense to apply too much analysis to it because it is not derived anywhere with rigour. As long as the corrollary is clear with the difference equation (rearranging helps) then I think that's all we really need to get our heads around.

Cheers
Michael

 

Hi all

I don't think the result is particularly "easily seen" either! That's why I added the explanation on Pages 39 and 40, which I'm hoping you're OK with.

Let's not worry too much about the bars over the V's and P's. They're just there to indicate that we are working in continuous time, with the premiums paid continuously and the death benefits paid immediately on death.

I agree with Michael that the last reserve term on the RHS should be (t+dt)V rather than tV. Thanks for pointing this out. I'll make sure that the change is made for the next edition of the notes.

However, the use of mu is OK. If you look at the third equation on Page 40 and multiply everything through by h (or dt if you're the Core Reading), you see that you have a term of the form [(t+h)V - S]* hq(x+t) on the RHS. Don't forget, though, that hq(x+t) = h*mu(x+t) + o(h), and the sum of a bunch of o(h) terms is another o(h) term.

Hope this helps

Julie

 

I don't see how (t+1)_V - t_V = i*t_V + P*(1+i) - q_(x+t)*(1 - t_V)

can be rearranged with a little algebra to be;

(t+1)_V - t_V = i*t_V + P*(1+i) - q_(x+t)*(1 - (t+1)_V)

also what happened to the p_x+t denominator?

The continous time version is worth a bit "belief" because of the o(dt) term.

Someone shine some light, thanks.

 

I'm not sure I do either. It's possible that "Michael_14" made a mistake - but then it was five years ago, so he's maybe passed the exam by now! His other reasoning seems OK to me, though.

What Core Reading is trying to get at is that the discrete recursive formula over one year and the continuous time version over a length of time dt are similar, or as "Louisa" says below, analogous. Further description of how to obtain Thiele's differential equation follows on the next couple of pages in the notes, so try to follow that through, as you get to the same place by general reasoning.

When moving from the discrete recursive formula to the continuous time version, the 1/p_x+t term would become 1/dt_p_x+t (ie one over the probability of surviving from age x+t for a short interval dt). We could argue that dt_p_x+t is very close to 1, so the 1/1 term disappears, and to the extent that 1/dt_p_x+t is not 1, the extra terms are swept up in o(dt).

 

Thanks,

But why is it that when moving from the discrete version to the continuous version, tVx becomes tVx with a bar on the L.H.S. However on the R.H.S the tVx multiplied by the probability of death becomes t+dtVx with a bar??