brownian.py

import numpy as np
from scipy.stats import norm

np.random.seed(0)

"""
brownian() implements one dimensional Brownian motion (i.e. the Wiener process).
"""

def brownian(x0, n, dt, delta, out=None):
    """
    Generate an instance of Brownian motion (i.e. the Wiener process):

        X(t) = X(0) + N(0, delta**2 * t; 0, t)

    where N(a,b; t0, t1) is a normally distributed random variable with mean a and
    variance b.  The parameters t0 and t1 make explicit the statistical
    independence of N on different time intervals; that is, if [t0, t1) and
    [t2, t3) are disjoint intervals, then N(a, b; t0, t1) and N(a, b; t2, t3)
    are independent.
    
    Written as an iteration scheme,

        X(t + dt) = X(t) + N(0, delta**2 * dt; t, t+dt)


    If `x0` is an array (or array-like), each value in `x0` is treated as
    an initial condition, and the value returned is a numpy array with one
    more dimension than `x0`.

    Arguments
    ---------
    x0 : float or numpy array (or something that can be converted to a numpy array
         using numpy.asarray(x0)).
        The initial condition(s) (i.e. position(s)) of the Brownian motion.
    n : int
        The number of steps to take.
    dt : float
        The time step.
    delta : float
        delta determines the "speed" of the Brownian motion.  The random variable
        of the position at time t, X(t), has a normal distribution whose mean is
        the position at time t=0 and whose variance is delta**2*t.
    out : numpy array or None
        If `out` is not None, it specifies the array in which to put the
        result.  If `out` is None, a new numpy array is created and returned.

    Returns
    -------
    A numpy array of floats with shape `x0.shape + (n,)`.
    
    Note that the initial value `x0` is not included in the returned array.
    """

    x0 = np.asarray(x0)

    # For each element of x0, generate a sample of n numbers from a
    # normal distribution.
    r = norm.rvs(size=x0.shape + (n,), scale=delta*np.sqrt(dt))

    # If `out` was not given, create an output array.
    if out is None:
        out = np.empty(r.shape)

    # This computes the Brownian motion by forming the cumulative sum of
    # the random samples. 
    np.cumsum(r, axis=-1, out=out)

    # Add the initial condition.
    out += np.expand_dims(x0, axis=-1)

    return out

# The Wiener process parameter.
delta = 2
# Total time.
T = 10.0
# Number of steps.
N = 500
# Time step size
dt = T/N
# Number of realizations to generate.
m = 20
# Create an empty array to store the realizations.
x = np.empty((m,N+1))
# Initial values of x.
x[:, 0] = 50