"""
BESQ Process
"""
from functools import partial
from multiprocessing import Pool
import numpy as np
from scipy.stats import ncx2
from aleatory.processes.analytical.brownian_motion import BrownianMotion
from aleatory.processes.base import SPExplicit
from aleatory.utils.utils import get_times, check_positive_integer, sample_besselq_global
def _sample_besselq_global(T, initial, dim, n):
path = sample_besselq_global(T=T, initial=initial, dim=dim, n=n)
return path
[docs]class BESQProcess(SPExplicit):
r"""Squared Bessel process
.. image:: _static/besq_process_drawn.png
A squared Bessel process :math:`BESQ^{n}_{0}`, for :math:`n` integer is a continuous stochastic process
:math:`\{X(t) : t \geq 0\}` which is characterised as the squared Euclidian norm of an :math:`n`-dimensional
Brownian motion. That is,
.. math::
X_t = \sum_{i=1}^n (W^i_t)^2.
More generally, for any :math:`\delta >0`, and :math:`x_0 \geq 0`, a squared Bessel process of
dimension :math:`\delta` starting at :math:`x_0`, denoted by
.. math::
BESQ_{{x_0}}^{{\delta}}
can be defined by the following SDE
.. math::
dX_t = \delta dt + 2\sqrt{X_t} dW_t \ \ \ \ t\in (0,T]
with initial condition :math:`X_0 = x_0`, where
- :math:`\delta` is a positive real
- :math:`W_t` is a standard Brownian Motion.
:param double dim: the dimension of the process :math:`n`
:param double initial: the initial point of the process :math:`x_0`
:param double T: the right hand endpoint of the time interval :math:`[0,T]`
for the process
:param numpy.random.Generator rng: a custom random number generator
"""
def __init__(self, dim=1.0, initial=0.0, T=1.0, rng=None):
super().__init__(T=T, rng=rng, initial=initial)
self.dim = dim
self._brownian_motion = BrownianMotion(T=T, rng=rng)
self.name = f'$BESQ^{{{self.dim}}}_{{{self.initial}}}$'
self.n = None
self.times = None
def __str__(self):
return "BESQ process with dimension {dim} and starting condition {initial} on [0, {T}].".format(
T=str(self.T), dim=str(self.dim), initial=str(self.initial))
def __repr__(self):
return "Squared Bessel Process(dimension={dim}, initial={initial}, T={T})".format(
T=str(self.T), dim=str(self.dim), initial=str(self.initial))
@property
def dim(self):
"""Bessel Process dimension."""
return self._dim
@dim.setter
def dim(self, value):
if value < 0:
raise TypeError("Dimension must be positive.")
self._dim = value
def _sample_besselq_alpha_integer(self, n):
check_positive_integer(n)
self.n = n
self.times = get_times(self.T, n)
brownian_samples = [self._brownian_motion.sample(n) for _ in range(self.dim)]
norm_squared = np.array([np.linalg.norm(coord) ** 2 for coord in zip(*brownian_samples)])
return norm_squared
def sample(self, n):
if isinstance(self.dim, int) and self.initial == 0:
return self._sample_besselq_alpha_integer(n)
else:
return _sample_besselq_global(self.T, self.initial, self.dim, n)
[docs] def simulate(self, n, N):
"""
Simulate paths/trajectories from the instanced stochastic process.
:param n: number of steps in each path
:param N: number of paths to simulate
:return: list with N paths (each one is a numpy array of size n)
"""
self.n = n
self.N = N
self.times = get_times(self.T, n)
if isinstance(self.dim, int) and self.initial == 0:
self.paths = [self.sample(n) for _ in range(N)]
return self.paths
else:
pool = Pool()
initial = self.initial
dim = self.dim
T = self.T
func = partial(_sample_besselq_global, T, initial, dim)
results = pool.map(func, [n] * N)
pool.close()
pool.join()
self.paths = results
return self.paths
def get_marginal(self, t):
marginal = ncx2(df=self.dim, nc=self.initial / t, scale=t)
# ncx2(df=dim, nc=x / t_size, scale=t_size).rvs(1)[0]
return marginal
def _process_expectation(self, times=None):
if times is None:
times = self.times
expectations = self.initial + self.dim * np.array(times)
return expectations
def marginal_expectation(self, times=None):
expectations = self._process_expectation(times=times)
return expectations
def _process_variance(self, times=None):
if times is None:
times = self.times
variances = 2.0 * (self.dim + 2.0*self.initial/times) * times**2
return variances
def marginal_variance(self, times):
variances = self._process_variance(times=times)
return variances
def _process_stds(self):
stds = np.sqrt(self._process_variance())
return stds
def process_stds(self):
stds = self._process_stds()
return stds