A gamma process, also called the Moran-Gamma subordinator, is a two-parameter stochastic process which models the accumulation of effort or wear over time. The gamma process has independent and stationary increments which follow the gamma distribution, hence the name. The gamma process is studied in mathematics, statistics, probability theory, and stochastics, with particular applications in deterioration modeling and mathematical finance.

Notation

controls the rate of jump arrivals, analogously with the gamma distribution. must be greater than 0. We use the gamma function and gamma distribution (the gamma process).

Definition

The process is a pure-jump increasing Lévy process occur as a Poisson process

and independent increments, whose marginal distribution is given by

Inhomogenous process

.

Properties

Mean and variance

.

Scaling

Adding independent processes

Moments

The moment function is the Gamma function.

Moment generating function

The moment generating function where X is the random variable

Correlation

Correlation

Related processes

The gamma process is used as the distribution for random time change in the variance gamma process. Specifically, combining Brownian motion with a gamma process produces a variance gamma process, and a variance gamma process can be written as the difference of two gamma processes.

See also

• Moran process

Notes

References