aleatory.processes.BESProcess

class aleatory.processes.BESProcess(dim=1.0, initial=0.0, T=1.0, rng=None)

Bessel process

Notes

A Bessel process \(BES^{n}_{0},\) for \(n\geq 2\) integer is a continuous stochastic process \(\{X(t) : t \geq 0\}\) characterised as the Euclidian norm of an \(n\)-dimensional Brownian motion. That is,

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

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

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

can be defined by the following SDE

\[dX_t = \frac{(\delta-1)}{2} \frac{dt}{X_t} + dW_t \ \ \ \ t\in (0,T]\]

with initial condition \(X_0 = x_0\geq 0.\), where

• \(\delta\) is a positive real

• \(W_t\) is a standard one-dimensional Brownian Motion.

Constructor, Methods, and Attributes

__init__(dim=1.0, initial=0.0, T=1.0, rng=None)

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_variance(times)
plot(n, N[, T, title, suptitle])	Simulates and plots paths/trajectories from the instanced stochastic process.
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.
initial	Bessel Process initial point.
rng