Approximate Comparison of Statistical Experiments with Filtered Probability Spaces

A statistical experiment with a filtered probability space is called a filtered experiment. The paper introduces a concept of deficiency for filtered decision problems of one filtered experiment with respect to another. The main theorem is a natural extension of Le Cam (1964, Thm. 3) and establishes a connection between the notion of 'deficiency' and several criteria for comparison of risks. The concept of deficiency developed here is relevant practical purposes of stochastic processes which depend on unknown parameters. The main result applies to statistical problems, like for instance, certain types of stochastic control problems, stopping time problems, and filtering problems