On the stochastic equation L(X)=L[B(X+C)] and a property of gamma distributions

Abstract

This paper is concerned with the stochastic equation X = B(X+C), where B, X and C are independent. This equation has appeared in a number of contexts, notably in actuarial science. An apparently new property of gamma variables (Theorem 1) leads to the derivation of a new explicit example of solution of the stochastic equation (Theorem 2), where B is the product of two independent beta variables, C is gamma and X is the product of independent beta and gamma variables. Also, a number of previously known explicit examples are seen to be direct algebraic consequences of a well-known property of gamma variables.