Gamma process

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A gamma process is a random process with independent gamma distributed increments. Often written as , it is a pure-jump increasing Lévy process with intensity measure , for positive . Thus jumps whose size lies in the interval occur as a Poisson process with intensity The parameter controls the rate of jump arrivals and the scaling parameter inversely controls the jump size. It is assumed that the process starts from a value 0 at t=0.

The gamma process is sometimes also parameterised in terms of the mean () and variance () of the increase per unit time, which is equivalent to and .

Properties[edit]

Since we use the Gamma function in these properties, we may write the process at time as to eliminate ambiguity.

Some basic properties of the gamma process are:[citation needed]

Marginal distribution[edit]

The marginal distribution of a gamma process at time is a gamma distribution with mean and variance

That is, its density is given by

Scaling[edit]

Multiplication of a gamma process by a scalar constant is again a gamma process with different mean increase rate.

Adding independent processes[edit]

The sum of two independent gamma processes is again a gamma process.

Moments[edit]

where is the Gamma function.

Moment generating function[edit]

Correlation[edit]

, for any gamma process

The gamma process is used as the distribution for random time change in the variance gamma process.

References[edit]

  • Lévy Processes and Stochastic Calculus by David Applebaum, CUP 2004, ISBN 0-521-83263-2.