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| In [[probability theory]], the '''telegraph process''' is a [[Memorylessness|memoryless]] continuous-time [[stochastic process]] that shows two distinct values.
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| If these are called ''a'' and ''b'', the process can be described by the following [[master equation]]s:
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| :<math>\partial_t P(a, t|x, t_0)=-\lambda P(a, t|x, t_0)+\mu P(b, t|x, t_0)</math>
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| and
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| :<math>\partial_t P(b, t|x, t_0)=\lambda P(a, t|x, t_0)-\mu P(b, t|x, t_0).</math>
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| The process is also known under the names [[Mark Kac|Kac]] process<ref name="Kac">{{cite journal | doi = 10.1023/A:1009437108439 | last1 = Bondarenko | first1 = YV | year = 2000 | title = Probabilistic Model for Description of Evolution of Financial Indices | url = | journal = Cybernetics and systems analysis | volume = 36 | issue = | pages = 738–742 }}</ref>
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| , dichotomous random process.<ref>{{cite journal | last1 = Margolin | first1 = G | last2 = Barkai | first2 = E | year = 2006 | title = Nonergodicity of a Time Series Obeying Lévy Statistics | url = | journal = Journal of Statistical Physics | volume = 122 | issue = | pages = 137–167 | doi =10.1007/s10955-005-8076-9 |bibcode=2006JSP...122..137M|arxiv = cond-mat/0504454 }}</ref>
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| ==Properties==
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| Knowledge of an initial state [[exponential decay|decays exponentially]]. Therefore for a time in the remote future, the process will reach the following stationary values, denoted by subscript ''s'':
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| Mean:
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| : <math>\langle X \rangle_s = \frac {a\mu+b\lambda}{\mu+\lambda}.</math>
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| Variance:
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| : <math> \operatorname{var} \{ X \}_s = \frac {(a-b)^2\mu\lambda}{(\mu+\lambda)^2}.</math>
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| One can also calculate a [[correlation function]]:
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| : <math>\langle X(t),X(s)\rangle_s = \exp(-(\lambda+\mu)|t-s|) \operatorname{var} \{ X \}_s.</math>
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| ==Application== | |
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| This random process finds wide application in [[model building]]:
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| * In [[physics]], [[Spin (physics)|spin systems]] and [[fluorescence]] [[fluorescence intermittency|intermittency]] show dichotomous properties. But especially in [[single molecule experiment]]s [[probability distribution]]s featuring [[algebraic tail]]s are used instead of the [[exponential distribution]] implied in all formulas above.
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| * In [[finance]] for describing [[stock]] prices<ref name="Kac" />
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| ==See also==
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| *[[Markov chain]]
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| *[[List of stochastic processes topics]]
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| *[[Random telegraph signal]]
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| ==References==
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| <references/>
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| {{Stochastic processes}}
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| [[Category:Stochastic differential equations]]
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| [[Category:Stochastic processes]]
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