aleatory.processes.BESQProcess#

class aleatory.processes.BESQProcess(dim=1.0, initial=0.0, T=1.0, rng=None)[source]#

Squared Bessel process#

../_images/besq_process_drawn.png

Definition#

A squared Bessel process \(BESQ^{n}_{0}\), for \(n\) integer is a continuous stochastic process \(\{X(t) : t \geq 0\}\) which is characterised as the squared Euclidian norm of an \(n\)-dimensional Brownian motion. That is,

\[X_t = \sum_{i=1}^n (W^i_t)^2.\]

More generally, for any \(\delta >0\), and \(x_0 \geq 0\), a squared Bessel process of dimension \(\delta\) starting at \(x_0\), denoted by

\[BESQ_{{x_0}}^{{\delta}}\]

can be defined by the following SDE

\[dX_t = \delta dt + 2\sqrt{X_t} dW_t \ \ \ \ t\in (0,T]\]

with initial condition \(X_0 = x_0\), where

  • \(\delta\) is a positive real

  • \(W_t\) is a standard Brownian Motion.

Constructor, Methods, and Attributes#

__init__(dim=1.0, initial=0.0, T=1.0, rng=None)[source]#
Parameters:

  • dim (double) – the dimension of the process \(n\)

  • initial (double) – the initial point of the process \(x_0\)

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

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

Methods

__init__([dim, initial, T, rng])

param double dim:

the dimension of the process \(n\)

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

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, title, suptitle])

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.

dim

Bessel Process dimension.

rng