Telegraph process

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In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values.

If these are called a and b, the process can be described by the following master equations:

\partial_t P(a, t|x, t_0)=-\lambda P(a, t|x, t_0)+\mu P(b, t|x, t_0)

and

\partial_t P(b, t|x, t_0)=\lambda P(a, t|x, t_0)-\mu P(b, t|x, t_0).

The process is also known under the names Kac process[1] , dichotomous random process.[2]

Properties[edit]

Knowledge of an initial state decays exponentially. Therefore for a time in the remote future, the process will reach the following stationary values, denoted by subscript s:

Mean:

\langle X \rangle_s = \frac {a\mu+b\lambda}{\mu+\lambda}.

Variance:

 \operatorname{var} \{ X \}_s = \frac {(a-b)^2\mu\lambda}{(\mu+\lambda)^2}.

One can also calculate a correlation function:

\langle X(t),X(s)\rangle_s = \exp(-(\lambda+\mu)|t-s|) \operatorname{var} \{ X \}_s.

Application[edit]

This random process finds wide application in model building:

See also[edit]

References[edit]

  1. ^ a b Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and systems analysis 36: 738–742. doi:10.1023/A:1009437108439. 
  2. ^ Margolin, G; Barkai, E (2006). "Nonergodicity of a Time Series Obeying Lévy Statistics". Journal of Statistical Physics 122: 137–167. arXiv:cond-mat/0504454. Bibcode:2006JSP...122..137M. doi:10.1007/s10955-005-8076-9.