Open Stochastic Systems

The problem of providing an adequate definition of a stochastic system is addressed and motivated using examples. A stochastic system is defined as a probability triple. The specification of the set of events is an essential part of a stochastic model and it is argued that for phenomena with as outcome space a finite dimensional vector space, the framework of classical random vectors with the Borel sigma-algebra as events is inadequate even for elementary applications. Models very often require a coarse event sigma-algebra. A stochastic system is linear if the events are cylinders with fibers parallel to a linear subspace of a vector space. We address interconnection of stochastic systems. Two stochastic systems can be interconnected if they are complementary. We discuss aspects of the identification problem from this vantage point. A notion that emerges is constrained probability, a concept that is reminiscent but distinct from conditional probability. We end up with a comparison of open stochastic systems with probability kernels.