Variance gamma process

In the theory of stochastic processes, a part of the mathematical theory of probability, the variance gamma (VG) 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 $W(t)$ with drift $\theta t$ subjected to a random time change which follows a gamma process $\Gamma (t;1,\nu )$ (equivalently one finds in literature the notation $\Gamma (t;\gamma =1/\nu ,\lambda =1/\nu )$ ):

X^{VG}(t;\sigma ,\nu ,\theta )\;:=\;\theta \,\Gamma (t;1,\nu )+\sigma \,W(\Gamma (t;1,\nu ))\quad .

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]

X^{VG}(t;\sigma ,\nu ,\theta )\;:=\;\Gamma (t;\mu _{p},\mu _{p}^{2}\,\nu )-\Gamma (t;\mu _{q},\mu _{q}^{2}\,\nu )

where

\mu _{p}:={\frac {1}{2}}{\sqrt {\theta ^{2}+{\frac {2\sigma ^{2}}{\nu }}}}+{\frac {\theta }{2}}\quad \quad {\text{and}}\quad \quad \mu _{q}:={\frac {1}{2}}{\sqrt {\theta ^{2}+{\frac {2\sigma ^{2}}{\nu }}}}-{\frac {\theta }{2}}\quad .

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]

^ Dilip Madan; Peter Carr; Eric Chang (1998). "The Variance Gamma Process and Option Pricing" (PDF). European FinanceReview. 2: 79–105.
^ 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. ISBN 978-0817641665.
^ 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. ISBN 978-0-8176-4544-1.

[MadanCarrChang-1] Dilip Madan; Peter Carr; Eric Chang (1998). "The Variance Gamma Process and Option Pricing" (PDF). European FinanceReview. 2: 79–105.

[sktkkp-2] 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. ISBN 978-0817641665.

[Seneta2000-3] 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. ISBN 978-0-8176-4544-1.

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