On stochastic recursive equations of sum- and max-type

In this paper we consider stochastic recursive equations of sumtype X d = ∑K i=1 AiXi+b and of max-type X d =max(AiXi+bi; 1 ≤ i ≤ k) where Ai, bi, b are random and (Xi) are iid copies of X. Equations of this type typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems as well as in many other problems having a recursive structure. We develop some new contraction properties of minimal Ls-metrics which allow to establish general existence and uniqueness results for solutions without posing any moment conditons. As application we obtain a one to one relationship between the set of solutions of the homogeneous equation and the set of solutions of the inhomogeneous equation for sumand max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler (2003) from nonnegative solutions of sum-type to those of max-type by means of random scaled Weibull distributions.