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. 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.

Basic properties

Every Gauss–Markov process X(t) possesses the three following properties:

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

Template:Main A stationary Gauss–Markov process with variance $\mathbf{\text{E}}(X^2(t))={\sigma }^2$ and time constant ${\beta }^{-1}$ has the following properties.

• Exponential autocorrelation: $${\mathbf{\text{R}}}_x(\tau )={\sigma }^2{e}^{-\beta |\tau |}.$$
• A power spectral density (PSD) function that has the same shape as the Cauchy distribution: $${\mathbf{\text{S}}}_x(j\omega )=\frac{2{\sigma }^2\beta }{{\omega }^2+{\beta }^2}.$$ (Note that the Cauchy distribution and this spectrum differ by scale factors.)
• The above yields the following spectral factorization:$${\mathbf{\text{S}}}_x(s)=\frac{2{\sigma }^2\beta }{-s^2+{\beta }^2}=\frac{\sqrt{2\beta }\sigma }{(s+\beta )}\cdot \frac{\sqrt{2\beta }\sigma }{(-s+\beta )}.$$ which is important in Wiener filtering and other areas.

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