3 state Markov processes

[tatos]

Consider a 3 state Markov Jump Process
H, Healthy S, Sick D, Dead
Instantaneous transition rates as follows:
H to S : a
S to H : b
H to D : c
S to D : d

the generator matrix is thus:

... ....H..... ....S.... .....D
H. -(a+c). ...a..... ....c
S. ....b..... -(b+d) ....d
D. ....0.... ....0.... .....0

If h is the time expected in state H before finally entering state D and s is the time expected in state S before finally entering state D, and if contributions are paid at rate C when in state H and benefits are received at rate B when in state S, and no death benefit is payable, then for solvency the company requires:

Ch - Bs >= 0

My problem is:

I don't understand how the following equations are derived

h = 1/(a+c) + [(a/(a+c))*s]

and

s = 1/(b+d) + [(b/(b+d)*h]

This is an adaptation of Question 7 (ii) in April 2003, subject 103.. but essentially the information is the same

 

[Mark Mitchell]

This formula (and the thinking behind it) is explained in Section 8 of Chapter 5. Why not have a read of that to see if it helps you? The end of chapter question 5.17 also uses this approach, so looking at that may also be of assistance.

Note that the formula underlying these equations is given on page 38 of the Tables, so you'll always have it to hand to remind you what to do.