Instantaneous state

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of a homogeneous Markov chain with a countable set of states

A state (say, $i$) for which the density of the transition probability,


where $p_{ii}(h)$ is the probability of transition from $i$ to $i$ in time $h$, is equal to $-\infty$. In the opposite case the state $i$ is called non-instantaneous, or retarded.


[1] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)



[a1] D. Williams, "Diffusions, Markov processes, and martingales" , 1 , Wiley (1979)
[a2] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967)
[a3] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian)
[a4] D. Freedman, "Brownian motion and diffusion" , Holden-Day (1971)
How to Cite This Entry:
Instantaneous state. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article