aleatory.processes.CIRProcess#

class aleatory.processes.CIRProcess(theta=1.0, mu=2.0, sigma=0.5, initial=5.0, T=1.0, rng=None)[source]#

Cox–Ingersoll–Ross (CIR) Process#

../_images/cir_process_drawn.png

Notes#

A Cox–Ingersoll–Ross process \(X = \{X : t \geq 0\}\) is characterised by the following Stochastic Differential Equation

\[dX_t = \theta(\mu - X_t) dt + \sigma \sqrt{X_t} dW_t, \ \ \ \ \forall t\in (0,T],\]

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

  • \(\theta\) is the rate of mean reversion

  • \(\mu\) is the long term mean value.

  • \(\sigma>0\) is the instantaneous volatility

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

It can be seen that each \(X_t\) follows a non-central chi-square distribution.

Constructor, Methods, and Attributes#

__init__(theta=1.0, mu=2.0, sigma=0.5, initial=5.0, T=1.0, rng=None)[source]#
Parameters:

  • theta (float) – the parameter \(\theta\) in the above SDE

  • mu (float) – the parameter \(\mu\) in the above SDE

  • sigma (float) – the parameter \(\sigma>0\) in the above SDE

  • initial (float) – the initial condition \(x_0\) in the above SDE

  • 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, mu, sigma, initial, T, rng])

param float theta:

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

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_df()

marginal_expectation([times])

marginal_nc_parameter([times])

marginal_scale([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.

mu

rng

sigma

theta