List of stochastic processes

The value of the following list lies in the classification of the processes in the various categories of processes. In addition, a comprehensive overview of the stochastic differential equations (SDGL) of the various processes and, if possible, their solutions should be created. The details are in the main articles of the respective processes.

List of stochastic processes .

Markov trials

Markov processes fulfill the Markov property. The Markov processes include a. the affine processes and the Itō processes.

Affine processes

Affine processes include a. the Lévy processes (including the Wiener process and the Poisson process), as well as some Itō processes such as B. the Ornstein-Uhlenbeck process and the root diffusion process.

Lévy trials

Lévy processes are processes with independent and stationary increments. The Lévy processes include a. the Poisson processes.

Gamma process

The gamma process is a pure jump Lévy process with a measure of intensity ${\displaystyle \Gamma (t;\gamma ,\lambda )}$ ${\displaystyle \nu (x)=\gamma x^{-1}\exp(-\lambda x)}$

Variance gamma process

${\displaystyle X(t;\theta ,\sigma ,\nu )=B(\Gamma (t;1,\nu ),\theta ,\sigma ),\qquad t\geq 0.}$

Poisson processes

Compound Poisson process

${\displaystyle X_{t}:=\sum _{n=1}^{N_{t}}Y_{n}}$

Inhomogeneous Poisson process

The intensity is time-dependent ${\displaystyle \lambda (t)}$

Spatial Poisson process

The intensity is time and (vector) space dependent ${\displaystyle \lambda (t,x)}$

Cox trial

The intensity is a random variable.

Itō trials

Itō process

${\displaystyle X_{t}=X_{0}+\int _{0}^{t}u(s,X_{s})ds+\int _{0}^{t}v(s,X_{s})dW_{s}.}$

SDGL:

${\displaystyle \mathrm {d} X_{t}=u(t,X_{t})dt+v(t,X_{t})dW_{t}.}$

Generalized Wiener process / Generalized Brownian motion

The generalized Wiener process is both Gauss and Ito processes.

${\displaystyle X_{t}=\int _{0}^{t}f(s)\mathrm {d} s+\int _{0}^{t}g(s)\mathrm {d} W_{s}}$

SDGL: ${\displaystyle \mathrm {d} X_{t}=f(t)\mathrm {d} t+g(t)\mathrm {d} W_{t}}$

Simple form

${\displaystyle X_{t}=\mu t+\sigma W_{t}}$

SDGL:

${\displaystyle \mathrm {d} X_{t}=\mu \mathrm {d} t+\sigma \mathrm {d} W_{t}}$

Standard Wiener process / Standard Brownian motion

${\displaystyle X_{t}=W_{t}=\int _{0}^{t}dW_{s}}$

SDGL:

${\displaystyle \mathrm {d} X_{t}=\mathrm {d} W_{t}}$

Other Itō trials

Geometric Brownian motion

${\displaystyle X_{t}=X_{0}e^{(\mu -{\frac {\sigma ^{2}}{2}})t+\sigma W_{t}}}$

SDGL:

${\displaystyle \mathrm {d} X_{t}=\mu X_{t}\mathrm {d} t+\sigma X_{t}\mathrm {d} W_{t}}$

Ornstein-Uhlenbeck trial

SDGL:

${\displaystyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dW_{t},\;\;X_{0}=a}$

Root diffusion process / CIR process

SDGL:

${\displaystyle dX_{t}=\kappa (\theta -X_{t})dt+\sigma {\sqrt {X_{t}}}dW_{t}}$

Bessel process

${\displaystyle X_{t}=\|W_{t}\|_{2},}$

SDGL:

${\displaystyle \mathrm {d} X_{t}=\mathrm {d} W_{t}+{\frac {n-1}{2}}{\frac {\mathrm {d} t}{X_{t}}}}$

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Gaussian processes

${\displaystyle (X_{t})_{t\in T}}$is a Gaussian process, if is is given by an n-dimensional normal distribution. ${\displaystyle \forall t_{1},t_{2}\ldots t_{n}\in T}$ ${\displaystyle (X_{t_{1}},X_{t_{2}}\ldots X_{t_{n}})}$

The Gaussian processes include a. the Gauss-Itō processes (e.g. the Wiener process), the Ornstein-Uhlenbeck process, the Brownian bridge, and the fractional Brownian motion.

Fractional Brownian Movement

Gauss-Markov trials

Gauss-Markow processes have both the Markov property and the property of Gaussian processes.

Brownian bridge

The Brownian Bridge is a Gauss-Markov process, i. H. a Gaussian process with the Markov property.

${\displaystyle B_{t}:=(W_{t}|W_{T}=0),\;t\in [0,T]}$

Feller trials

A Feller process is a Markov process with the Feller transition function that belongs to a Feller semigroup. The Feller processes include a. the Lévy processes, the Bessel process, and the SDGL solutions with Lipschitz continuous coefficients.