Note that [D.sub.f] has two

absorbing states corresponding to success and failure of execution, respectively.

In the HMF setting, under PI dynamics, when [epsilon] [member of] (0,1) the final state for the population is the

absorbing state [rho] =0 (full defection) when [epsilon] < c, [rho] = [rho] (0) when [epsilon] = c, and [rho] = 1 when [epsilon] > c.

[T.sub.w] includes [T.sub.S(1,0)[right arrow]s(w, .)], which is the transition time from initial state to any state with VM = w (and A U may be 0, 1, 2, ..., w); [[DELTA].sub.t,w], which stands for the time needed for a state with VM = w to evolve into the

absorbing state and to w, the RTO at window size.

Raja Nassar, Louisiana Tech University Table 1: Definitions of the different states of the Markov chain Transient States:

Absorbing States: Past Due and Prepayment States Default States [R.sub.k] [S.sub.i]:i = -3, -2, -1, 0, 1, 2, 3 [R.sub.k]:k = 1, 2, 3, 4 [S.sub.-3] Prepaid more than 90 days [R.sub.1] Sold by Bank [S.sub.-2] Prepaid 61 days--90 days [R.sub.2] Foreclosure [S.sub.-1] Prepaid 31 days--60 days [R.sub.3] Refuse to pay [S.sub.0] No more than 30 days [R.sub.4] All others past due reasons [S.sub.1] 31 days--60 days past due [S.sub.2] 61 days--90 days past due [S.sub.3] More than 90 days past due Table 2: [N.sub.t], the total number of retail mortgages in the transient states at time t.

Also, let [b.sub.ik] be the probability that the process transits from transient state i, i =-3,-2,1,0,1,2,3 to

absorbing state k, k=1,2,3,4:

Clearly, unemployment has become a much less

absorbing state for the whole working-age population, and this is particularly true for the younger age group.

Entities can be modelled in such a way that they leave the system without having to resort to the artificial device of an (Markov)

absorbing state. More importantly, it is easy to model entities entering the system so it becomes possible to model the behaviour of the overall system over time rather than just the particular cohort that was in the initial state.

For all possible values of the decision variables [[delta].sub.ij], the chain always exhibits

absorbing states. Specifically, a triple ([X.sub.1], [X.sub.2], [X.sub.3]) specifying the states of each individual can only be an

absorbing state if [X.sub.i] [member of] [S,S',R,V} for i = 1,2,3, because there is always a nonzero transition rate from any state containing an individual in states E, E',or I (Figure 1).

The r x r identity matrix I corresponds to transitions between

absorbing states. 0 is an r x (m - r) zero matrix, because transitions from

absorbing states to transient states are not possible.

Each row of Exhibit 8 represents the transient states, and each column represents the

absorbing states. Notice that a receivable in month 0 has an 89% probability of collection and an 11% probability of write-off.