Multi-state models are an attempt to look at a variety of life insurance and annuity contracts in a unified manner, by making use of Markov processes. As motivation, take an individual now age *x* and consider a two-state Markov chain, where the person is in state 0 (alive) or state 1 (deceased) at any time. Life insurance contracts provide benefits upon transfer from state 0 to state 1, while life annuity contracts provide benefits as long as the process remains in state 0.

More generally, consider a multiple-decrement model for (*x*) with *m* causes of failure. We can consider a chain with *m* + 1 states. State 0 means that (*x*) has not succumbed to any cause and is often referred to as the *active* state. State *j* refers to having succumbed first to cause *j*. The insurance benefits discussed in Chapter 11 can be viewed as payments upon transfer from state 0 to other states.

For still another example, consider a joint-life contract issued to (*x*) and (*y*). We can now take a chain with four states as illustrated in Figure 19.1. The arrows indicate that there are possible transitions from state 0 into state 1 or state 2, occasioned by the death of (*y*) or (*x*) respectively, and then further transitions from state 1 or state 2 into state 3 when the second death occurs. The dotted line, showing a transition directly from state 0 to state 3, would not be present in our original model, but would be there if we wanted to consider a common shock model as discussed ...

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