Abstract
This chapter introduces rigorously the most general of stochastic hybrid systems. The model structure is based on a typical hybrid automaton where all the elements are randomised. Concretely, the continuous evolution is described by specific stochastic differential equations and the discrete evolution is governed by specific stochastic kernels. Necessary assumptions regarding the existence of solutions and non-Zeno executions are also imposed. The resulting stochastic hybrid process can be thought of as a concatenation of diffusion processes where the switching mechanism is given by a Markov jump process. For the analysis of such a process an entire mathematical arsenal is required. The proof of Markov property of the stochastic hybrid process, its evolution is analysed from the perspective of sequential composition of Markov processes. The critical point is how to choose the jumping times. For preserving the Markov property, it is necessary that the jumping times to have memoryless property, so they might be exponentially distributed or some appropriate hitting times (corresponding to the guard sets).
Analytic tools that are available for stochastic hybrid systems (infinitesimal generator, martingale problem) are presented and discussed at the end of this chapter. Furthermore, special features of the infinitesimal generator are underlined and discussed.
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- 1.
In stochastic analysis a killed process is a stochastic process that is forced to assume an undefined or ‘killed’ state at some (possibly random) time.
- 2.
Following [64], f is in \(L_{1}^{\mathit{loc}}(p)\) if for some sequence of stopping times σ n ↑∞
$$\begin{aligned} E_{x}\sum_{i}\big|f(x_{T_{i}\wedge \sigma _{n}})-f(x_{T_{i}\wedge \sigma _{n}-})\big|< \infty . \end{aligned}$$
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Bujorianu, L.M. (2012). Stochastic Hybrid Systems. In: Stochastic Reachability Analysis of Hybrid Systems. Communications and Control Engineering. Springer, London. https://doi.org/10.1007/978-1-4471-2795-6_4
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