  In this paper we propose a compositional framework for the construction of
approximations of the interconnection of a class of stochastic hybrid systems.
As special cases, this class of systems includes both jump linear stochastic
systems and linear stochastic hybrid automata. In the proposed framework, an
approximation is itself a stochastic hybrid system, which can be used as a
replacement of the original stochastic hybrid system in a controller design
process. We employ a notion of so-called stochastic simulation function to
quantify the error between the approximation and the original system. In the
first part of the paper, we derive sufficient conditions which facilitate the
compositional quantification of the error between the interconnection of
stochastic hybrid subsystems and that of their approximations using the
quantified error between the stochastic hybrid subsystems and their
corresponding approximations. In particular, we show how to construct
stochastic simulation functions for approximations of interconnected stochastic
hybrid systems using the stochastic simulation function for the approximation
of each component. In the second part of the paper, we focus on a specific
class of stochastic hybrid systems, namely, jump linear stochastic systems, and
propose a constructive scheme to determine approximations together with their
stochastic simulation functions for this class of systems. Finally, we
illustrate the effectiveness of the proposed results by constructing an
approximation of the interconnection of four jump linear stochastic subsystems
in a compositional way.
