In life insurance and finance, stochastically discounted sums of the following type are important: $${S_n}: = \sum\limits_{k = 1}^n {{Z_1} \cdots {Z_k}{Y_k}} .{\text{ }}(n = 1, \ldots ,\infty ) $$ where (Z i) and (Vi) are independent iid sequences satisfying mild moment conditions. Sums of this type are called perpetuities. The Y i are interpreted as payments, the Zi as discount factors. The distributional properties of S n, for n < ∞, can be studied via the following random equation: $${S_n}\mathop = \limits^{\text{d}} \left( {{Y_1} + {S_{n - 1}}} \right){Z_1}\left( {n = 1,2, \ldots } \right),$$ where S o:= 0. In this paper we discuss some aspects of the behaviour of solutions of this and related random equations, and their applications to perpetuities.
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