  Contemporary insurance theory is concentrated on models with different types
of polices and shock events may influence the payments on some of them.
Jordanova (2018) considered a model where a shock event contributes to the
total claim amount with one and the same value of the claim sizes to different
types of polices. Jordanova and Veleva (2021) went a step closer to real-life
situations and allowed a shock event to cause different claim sizes to
different types of polices. In that paper, the counting process is assumed to
be Multinomial. Here it is replaced with different independent homogeneous
Poison processes. The bivariate claim counting process is expressed in two
different ways. Its marginals and conditional distributions are totally
described. The mean square regression of these processes is computed. The
Laplace-Stieltjes transforms and numerical characteristics of the total claim
amount processes are obtained. The risk reserve process and the probabilities
of ruin in infinite time are discussed. The risk reserve just before the ruin
and the deficit (or the severity) at ruin are thoroughly investigated in the
case when the initial capital is zero. Their means, probability mass functions,
and probability generating functions are obtained.
  Although the model is constructed by a multivariate counting process, along
the paper it is shown that the total claim amount process is stochastically
equivalent to a univariate compound Poisson process. These allow us to reduce
the considered risk model to a Cramer-Lundberg risk model, to use the
corresponding results, and to make the conclusions for the new model. Analogous
results can be obtained for more types of polices and more types of shock
events.
  The results are applied in case when the claim sizes are exponentially
distributed.
  Stochastically equivalent models could be analogously constructed in queuing
theory.
