Analytical Processes#

The aleatory.processes module provides classes for continuous-time stochastic processes which can be expressed in analytical form.

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

Brownian motion

_images/brownian_motion_drawn.png

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

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

  • scale (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

property T#

End time of the process.

draw(n, N, marginal=True, envelope=False, type='3sigma', **fig_kw)[source]#

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

Produces different kind of visualisation illustrating the following elements:

  • times versus process values as lines

  • the expectation of the process across time

  • histogram showing the empirical marginal distribution \(X_T\) (optional when marginal = True)

  • probability density function of the marginal distribution \(X_T\) (optional when marginal = True)

  • envelope of confidence intervals acroos time (optional when envelope = True)

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

  • marginal – bool, default: True

  • envelope – bool, default: False

  • type – string, default: ‘3sigma’

Returns:

plot(n, N, **fig_kw)#

Simulates and plots paths/trajectories from the instanced stochastic process. Simple plot of times versus process values as lines and/or markers.

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

Returns:

sample(n)[source]#

Generates a discrete time sample from a Brownian Motion instance.

Parameters:

n – the number of steps

Returns:

numpy array

sample_at(times)[source]#

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

Parameters:

times – the times which define the sample

Returns:

numpy array

simulate(n, N)#

Simulate paths/trajectories from the instanced stochastic process.

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

Returns:

list with N paths (each one is a np.array of size n)

class aleatory.processes.GBM(drift=1.0, volatility=0.5, initial=1.0, T=1.0, rng=None)[source]#

Geometric Brownian Motion

_images/geometric_brownian_motion_drawn.png

A Geometric Brownian Motion \(\{X(t) : t \geq 0\}\) is characterised by the following SDE.

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

with initial condition \(X_0 = x_0\geq0\), where

  • \(\mu\) is the drift

  • \(\sigma>0\) is the volatility

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

The solution to this equation can be written as

\[X_t = x_0\exp\left((\mu + \frac{\sigma^2}{2} )t +\sigma W_t\right)\]

and each \(X_t\) follows a log-normal distribution.

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

  • volatility (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

property T#

End time of the process.

draw(n, N, marginal=True, envelope=False, **fig_kw)#

Simulates and plots paths/trajectories from the instanced stochastic process. Visualisation shows - times versus process values as lines - the expectation of the process across time - histogram showing the empirical marginal distribution \(X_T\) - probability density function of the marginal distribution \(X_T\) - envelope of confidence intervals

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

  • marginal – bool, default: True

  • envelope – bool, default: False

Returns:

plot(n, N, **fig_kw)#

Simulates and plots paths/trajectories from the instanced stochastic process. Simple plot of times versus process values as lines and/or markers.

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

Returns:

sample(n)[source]#

Generate a realization.

sample_at(times)[source]#

Generate a realization using specified times.

simulate(n, N)#

Simulate paths/trajectories from the instanced stochastic process.

Parameters:
  • n – number of steps in each path

  • N – number of paths to simulate

Returns:

list with N paths (each one is a np.array of size n)