Variance gamma process

In the theory of stochastic processes, 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. In the theory of stochastic processes, 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 W ( t ) {displaystyle W(t)} with drift θ t {displaystyle heta t} subjected to a random time change which follows a gamma process Γ ( t ; 1 , ν ) {displaystyle Gamma (t;1, u )} (equivalently one finds in literature the notation Γ ( t ; γ = 1 / ν , λ = 1 / ν ) {displaystyle Gamma (t;gamma =1/ u ,lambda =1/ u )} ): 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: