Modern Stochastics: Theory and Applications

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process S under a probability measure P we characterize all probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound renewal process under Q. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

This note gathers what is known about, and provides some new results concerning the operations of intersection, of “generated σ-field”, and of “complementation” for (independent) complete σ-fields on probability spaces.