EM Processes

The aleatory.processes module also provides functionality for continuous-time stochastic processes which cannot be expressed in analytical form. These processes are simulated via the Euler-Maruyama method.

• aleatory.processes.OUProcess

• aleatory.processes.Vasicek

• aleatory.processes.CIRProcess

• aleatory.processes.CEVProcess

class aleatory.processes.OUProcess(theta=1.0, sigma=0.5, initial=1.0, T=1.0, rng=None)

Ornstein–Uhlenbeck Process

An Ornstein–Uhlenbeck process \(X = \{X : t \geq 0\}\) is characterised by the following Stochastic Differential Equation

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

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

• \(\theta\) is the speed of reversion

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

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

Each \(X_t\) follows a normal distribution.

Parameters:

• theta (float) – the parameter \(\theta\) 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

property T

End time of the process.

draw(n, N, marginal=True, envelope=False, title=None, **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

• title – string optional default to None

Returns:

plot(n, N, title=None, **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

• title – string to customise plot title

Returns:

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 numpy array of size n)

class aleatory.processes.Vasicek(theta=1.0, mu=3.0, sigma=0.5, initial=1.0, T=1.0, rng=None)

Vasicek Process

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

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

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

• \(\theta\) is the speed of reversion

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

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

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

Each \(X_t\) follows a normal distribution.

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

property T

End time of the process.

draw(n, N, marginal=True, envelope=False, title=None, **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

• title – string optional default to None

Returns:

plot(n, N, title=None, **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

• title – string to customise plot title

Returns:

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 numpy array of size n)

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

Cox–Ingersoll–Ross Process

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.

Each \(X_t\) follows a non-central chi-square distribution.

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

property T

End time of the process.

draw(n, N, marginal=True, envelope=False, title=None, **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

• title – string optional default to None

Returns:

plot(n, N, title=None, **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

• title – string to customise plot title

Returns:

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 numpy array of size n)

class aleatory.processes.CEVProcess(gamma=1.5, mu=0.5, sigma=0.1, initial=1.0, T=1.0, rng=None)

CEV or constant elasticity of variance process

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

\[dX_t = \mu X_t dt + \sigma X_t^{\gamma} dW_t, \ \ \ \ \forall t\in (0,T],\]

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

• \(\mu\) is the drift

• \(\sigma>0\) is the scale of the volatility

• \(\gamma\geq 0\) is the elasticity term

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

Parameters:

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

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

• gamma (float) – the parameter \(\gamma\) 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, title=None, **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

• title – string optional default to None

Returns:

plot(n, N, title=None, **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

• title – string to customise plot title

Returns:

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 numpy array of size n)