There are several representations of the VG process that relate it to other processes. It can for example be written as a Brownian motion with drift subjected to a random time change which follows a gamma process (equivalently one finds in literature the notation ):
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:[1]
where
Alternatively it can be approximated by a compound Poisson process that leads to a representation with explicitly given (independent) jumps and their locations. This last characterization gives an understanding of the structure of the sample path with location and sizes of jumps.[2]
On the early history of the variance-gamma process see Seneta (2000).[3]
^Kotz, Samuel; Kozubowski, Tomasz J.; Podgórski, Krzysztof (2001). The Laplace distribution and generalizations : a revisit with applications to communications, economics, engineering, and finance. Boston [u.a.]: Birkhäuser. ISBN978-0817641665.
^Eugene Seneta (2000). "The Early Years of the Variance–Gamma Process". In Michael C. Fu; Robert A. Jarrow; Ju-Yi J. Yen; Robert J. Elliott (eds.). Advances in Mathematical Finance. Boston: Birkhauser. ISBN978-0-8176-4544-1.