On the solution to numerical problems using stochastic arithmetic

We investigate some algebraic properties of the system of stochastic numbers with the arithmetic operations addition and multiplication by scalars and the relation inclusion and point out certain practically important consequences from these properties. Our idea is to start from a minimal set of empirically known properties and to study these properties by an axiomatic approach. Based on this approach we develop an algebraic theory of stochastic numbers. A numerical example based on the Lagrange polynomial demonstrates the consistency between the CESTAC method and the presented theory of stochastic numbers.