aleatory.processes.VarianceGammaProcess#

class aleatory.processes.VarianceGammaProcess(theta=0.0, nu=1.0, sigma=1.0, T=1.0, rng=None)[source]#

Variance Gamma Process#

../_images/variance_gamma_draw.png

Notes#

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 Variance-Gamma process that relate it to other processes. It can for example be written as a Brownian motion \(W\) with drift \(\theta t\) subjected to a random time change which follows Gamma process \(\Gamma(t;1 ,\nu)\). That is,

\[X(t; \sigma, \nu, \theta) = \theta \Gamma(t;1, \nu) + \sigma W(\Gamma(t;1, \nu)); \quad t \in [0,T].\]

Constructor, Methods, and Attributes#

__init__(theta=0.0, nu=1.0, sigma=1.0, T=1.0, rng=None)[source]#
Parameters:

  • theta (double) – the \(\theta\) parameter in the above expression

  • nu (double) – the \(\nu\) parameter in the above expression

  • sigma (double) – the \(\sigma\) parameter in the above expression

  • T (float) – the right hand endpoint of the time interval \([0,T]\) for the process

  • rng (numpy.random.Generator) – a custom random number generator

Methods

__init__([theta, nu, sigma, T, rng])

parameter double theta:

the \(\theta\) parameter in the above expression

draw(n, N[, T, mode, marginal, envelope, ...])

Simulates and plots paths/trajectories from the instanced stochastic process.

estimate_covariances([times])

estimate_expectations()

estimate_quantiles(q)

estimate_stds()

estimate_variances()

get_marginal(t)

marginal_expectation([times])

marginal_stds([times])

marginal_variance([times])

plot(n, N[, T, mode, title])

Simulates and plots paths/trajectories from the instanced stochastic process.

plot_covariance([times, title])

plot_kernel([times, colormap, matrix_shape, ...])

plot_kernel3d([times, title])

plot_mean_variance([times, title])

plot_paths_and_kernel(n, N[, T, title, ...])

Plots the paths of the process and the covariance kernel.

process_covariance([times])

process_expectation()

process_stds()

process_variance()

sample(n)

simulate(n, N[, T])

Simulate paths/trajectories from the instanced stochastic process.

Attributes

T

End time of the process.

rng