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In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma process , 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 W {\displaystyle W} with drift θ t {\displaystyle \theta t} subjected to a random time change which follows a gamma process Γ {\displaystyle \Gamma } {\displaystyle \Gamma } ]:
An alternative way of stating this is that the variance gamma process is a Brownian motion subordinated to a gamma subordinator.
Since the VG process is of finite variation it can be written as the difference of two independent gamma processes: