Samiksha Jaiswal (Editor)

Telegraph process

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

Contents

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

∂ t P ( a , t | x , t 0 ) = − λ P ( a , t | x , t 0 ) + μ P ( b , t | x , t 0 )

and

∂ t P ( b , t | x , t 0 ) = λ P ( a , t | x , t 0 ) − μ P ( b , t | x , t 0 ) .

The process is also known under the names Kac process , dichotomous random process.

Properties

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:

⟨ X ⟩ s = a μ + b λ μ + λ .

Variance:

var ⁡ { X } s = ( a − b ) 2 μ λ ( μ + λ ) 2 .

One can also calculate a correlation function:

⟨ X ( t ) , X ( s ) ⟩ s = exp ⁡ ( − ( λ + μ ) | t − s | ) var ⁡ { X } s .

Application

This random process finds wide application in model building:

  • In physics, spin systems and fluorescence intermittency show dichotomous properties. But especially in single molecule experiments probability distributions featuring algebraic tails are used instead of the exponential distribution implied in all formulas above.
  • In finance for describing stock prices

    • References

    Telegraph process Wikipedia
    (Text) CC BY-SA


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