In this paper, we consider hybrid containing both stochastic and deterministic (¹)  components. To compose such systems, we introduce a general combinator which allows the specification of an arbitrary hybrid system in terms of elementary components of only two types. Thus, systems are obtained hierarchically, by composing subsystems, where each subsystem can be viewed as an ``increment'' in the decomposition of the full system. The resulting hybrid stochastic system specifications are generally not ``executable'', since they do not necessarily permit the incremental simulation of the system variables. Such a simulation requires compiling the dependency relations existing between the system variables. Another issue involves finding the most likely internal states of a stochastic system from a set of observations. We provide a small set of primitives for transforming hybrid systems, which allows the solution of the two problems of incremental simulation and estimation of stochastic systems within a common framework. The complete model is called CSS (a Calculus of Stochastic Systems). Our results are applicable to pattern recognition problems formulated in terms of Markov random fields or Hidden Markov models (HMMs), and to the automatic generation of diagnostic systems for industrial plants
starting from their risk analysis.

Keywords : stochastic systems, hybrid systems, belief functions, communicating processes, simulation, estimation.

(¹) Throughout this paper, we use the word ``deterministic'' to refer to systems which have no random part. In control science or statistics, such systems would be called ``deterministic'' as opposed to ``stochastic'' ; however this name would be misleading in computer science, where ``deterministic'' vs. ``nondeterministic'' has a totally different meaning. This is why we decided to use the word ``deterministic'' here.