Variance gamma process

In the theory of , a part of the mathematical theory of probability, the variance gamma process (VG), also known as Laplace motion, is a Lévy process determined by a random time change. The process has finite moments distinguishing it from many Lévy processes. There is no diffusion component in the VG process and it is thus a pure jump process. The increments are independent and follow a Variance-gamma distribution, which is a generalization of the Laplace distribution.

There are several representations of the VG process that relate it to other processes. It can for example be written as a Brownian motion ${\displaystyle W(t)}$ with drift ${\displaystyle \theta t}$ subjected to a random time change which follows a gamma process ${\displaystyle \Gamma (t;1,\nu )}$ (equivalently one finds in literature the notation ${\displaystyle \Gamma (t;\gamma =1/\nu ,\lambda =1/\nu )}$):

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