Gauss–Markov process

Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes.<ref name=Rasmussen2006>Template:Cite book</ref><ref name=Lamon2008>Template:Cite book</ref> A stationary Gauss–Markov process is uniqueTemplate:Citation needed up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.

Gauss–Markov processes obey Langevin equations.<ref>Template:Cite book</ref>

Basic properties

Every Gauss–Markov process X(t) possesses the three following properties:<ref> C. B. Mehr and J. A. McFadden. Certain Properties of Gaussian Processes and Their First-Passage Times. Journal of the Royal Statistical Society. Series B (Methodological), Vol. 27, No. 3(1965), pp. 505-522</ref>

1. If h(t) is a non-zero scalar function of t, then Z(t) = h(t)X(t) is also a Gauss–Markov process
2. If f(t) is a non-decreasing scalar function of t, then Z(t) = X(f(t)) is also a Gauss–Markov process
3. If the process is non-degenerate and mean-square continuous, then there exists a non-zero scalar function h(t) and a strictly increasing scalar function f(t) such that X(t) = h(t)W(f(t)), where W(t) is the standard Wiener process.

Property (3) means that every non-degenerate mean-square continuous Gauss–Markov process can be synthesized from the standard Wiener process (SWP).

Other properties

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A stationary Gauss–Markov process with variance <math>\textbf{E}(X^{2}(t)) = \sigma^{2}</math> and time constant <math>\beta^{-1}</math> has the following properties.

• Exponential autocorrelation: <math display="block">\textbf{R}_{x}(\tau) = \sigma^{2}e^{-\beta |\tau|}.</math>
• A power spectral density (PSD) function that has the same shape as the Cauchy distribution: <math display="block"> \textbf{S}_{x}(j\omega) = \frac{2\sigma^{2}\beta}{\omega^{2} + \beta^{2}}.</math> (Note that the Cauchy distribution and this spectrum differ by scale factors.)
• The above yields the following spectral factorization:<math display="block">\textbf{S}_{x}(s) = \frac{2\sigma^{2}\beta}{-s^{2} + \beta^{2}}

= \frac{\sqrt{2\beta}\,\sigma}{(s + \beta)} 
  \cdot\frac{\sqrt{2\beta}\,\sigma}{(-s + \beta)}. </math> which is important in Wiener filtering and other areas.

There are also some trivial exceptions to all of the above.Template:Clarify