aleatory.processes.BrownianMeander#

class aleatory.processes.BrownianMeander(T=1.0, fixed_end=False, end=None, rng=None)[source]#

Brownian Meander#

../_images/brownian_meander_drawn.png ../_images/tied_brownian_meander_drawn.png

Notes#

Let \(\{W_t : t \geq 0\}\) be a standard Brownian motion and

\[\tau = \sup\{ t \in [0,1] : W_t =0\},\]

i.e. the last time before t = 1 when \(W_t\) visits zero. Then the Brownian Meander is defined as follows

\[W_t^{+} = \frac{1}{\sqrt{1-\tau}} |W(\tau + t (1-\tau))|, \ \ \ \ t\in (0,1].\]

Constructor, Methods, and Attributes#

__init__(T=1.0, fixed_end=False, end=None, rng=None)[source]#
Parameters:

  • T (float) – the right hand endpoint of the time interval \([0,T]\) for the process

  • fixed_end (bool) – flag to indicate if the process has a fixed end point. Defaults to False

  • end (float) – end point for the Meander, in the case of fixed_end equal True

  • rng (numpy.random.Generator) – a custom random number generator

Methods

__init__([T, fixed_end, end, rng])

param float T:

the right hand endpoint of the time interval \([0,T]\) for the process

draw(n, N[, title, suptitle])

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

estimate_covariances([times])

estimate_expectations()

estimate_quantiles(q)

estimate_stds()

estimate_variances()

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)

sample_at(times)

simulate(n, N[, T])

Simulate paths/trajectories from the instanced stochastic process.

Attributes

T

End time of the process.

end

rng