Characterization of the marginal distributions of Markov processes used in dynamic reliability. (English) Zbl 1101.60068

Consider a two-component Markov process \((I_t,X_t)\) with state space \(E \times \mathbb{R}^d\), \(E\) finite, and \(dX_t(\omega)/dt=v(i,X_ (\omega))\), given \(I_t (\omega)=i\). The marginal distribution of the process at time \(t\) is shown to be the unique measure solution of a set of integro-differential equations which may be viewed as a weak form of the Chapman-Kolmogorov equation.


60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J25 Continuous-time Markov processes on general state spaces
93Exx Stochastic systems and control
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