aleatory.processes.BrownianMotion#

class aleatory.processes.BrownianMotion(drift=0.0, scale=1.0, initial=0.0, T=1.0, rng=None)[source]#

Brownian Motion#

A one-dimensional standard Brownian motion object.

../_images/brownian_motion_drawn.png

Notes#

A standard Brownian motion \(\{W_t : t \geq 0\}\) is defined by the following properties:

  1. Starts at zero, i.e. \(W(0) = 0\)

  2. Independent increments

  3. \(W(t) - W(s)\) follows a Gaussian distribution \(N(0, t-s)\)

  4. Almost surely continuous

A more general version of a Brownian motion, is the Arithmetic Brownian Motion which is defined by the following SDE

\[dX_t = \mu dt + \sigma dW_t \ \ \ \ t\in (0,T]\]

with initial condition \(X_0 = x_0\in\mathbb{R}\), where

  • \(\mu\) is the drift

  • \(\sigma>0\) is the volatility

  • \(W_t\) is a standard Brownian Motion

Clearly, the solution to this equation can be written as

\[X_t = x_0 + \mu t + \sigma W_t \ \ \ \ t \in [0,T]\]

and each \(X_t \sim N(\mu t, \sigma^2 t)\).

Examples#

from aleatory.processes import BrownianMotion
process = BrownianMotion()
fig = process.plot(n=100, N=5, figsize=(12, 7))
fig.show()

from aleatory.processes import BrownianMotion
process = BrownianMotion()
fig = process.draw(n=100, N=100, figsize=(12, 7))
fig.show()

Constructor, Methods, and Attributes#

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

  • drift (double) – the drift parameter \(\mu\) in the above SDE

  • scale (double) – the scale parameter \(\sigma\) in the above SDE

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

  • T (double) – the endpoint of the time interval \([0,T]\) over which the process is defined

  • rng – random number generator for reproducibility

Methods

__init__([drift, scale, initial, T, rng])

param double drift:

the drift parameter \(\mu\) in the above SDE

draw(n, N[, T, marginal, envelope, type, 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, **fig_kw)

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)

Generates a discrete time sample from a Brownian Motion instance.

sample_at(times)

Generates a sample from a Brownian motion at the specified times.

simulate(n, N[, T])

Simulate paths/trajectories from the instanced stochastic process.

Attributes

T

End time of the process.

drift

initial

rng

scale