Stochastic Arithmetic : addition and multiplication by scalars

International audienceStochastic arithmetic involving addition and multiplication by scalars is studied with an emphasis on the abstract structure of the set of stochastic numbers. New properties of stochastic numbers are obtained such as a special distributivity relation corresponding to the second distributivity law in a vector space. This allows us to introduce algebraic systems abstracting properties of stochastic numbers, with respect to addition and multiplication by scalars. We define axiomatically such systems with group structure and give them a complete characterization in the finite dimensional case. This permits to reduce computation with stochastic numbers to computation in familiar vector spaces