In this chapter we deal with a perfectly general failure time *T*, and develop the stochastic approach for calculating premiums and reserves for insurance and annuity contracts based on *T*. These will all be calculated as expected values of appropriate random variables. In the particular case where *T* = *T*(*x*), we will show, using the correspondence established in Chapter 15, that these agree with the results that we obtained in the deterministic model. The advantage of the stochastic approach is that we can augment these expected values with other quantities, such as variances.

Throughout the chapter, *f*, *s* and μ will denote the density, survival and hazard functions respectively of *T*. We let be the associated discrete failure time, ⌊*T*⌋ + 1. We will let and λ denote the probability function and hazard function respectively of . We denote the survival function of with the same symbol *s*. This will not cause confusion since the value at any integer is the same in both cases.

In some cases the values of *T* are bounded ...

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