Exploration
Sampling Brownian Motion
From independent Gaussian increments to a sample path, with a small NumPy implementation.
The following code shows how we can sample a Brownian motion. As a reminder, a stochastic process \(\{B(t), t \geq 0\}\) is a standard Brownian motion if it satisfies the following properties:
- \(B(0) = 0\) almost surely.
- The process has continuous sample paths.
- The increments are stationary and independent on disjoint time intervals.
- \(B(t) - B(s)\) has a normal distribution with mean \(0\) and variance \(t-s\) for all \(0 \leq s < t\).
A discrete sample path
We sample independent normal increments with standard deviation \(\sqrt{\Delta t}\), then take their cumulative sum. The class below also supports a drift \(\mu\) and scale \(\sigma\), giving \(X(t) = \mu t + \sigma B(t)\). The defaults give standard Brownian motion.
import numpy as np
class BrownianMotion:
def __init__(self, mu=0.0, sigma=1.0, seed=None):
if not np.isfinite(mu) or not np.isfinite(sigma) or sigma < 0:
raise ValueError("Use a finite drift and a nonnegative finite scale.")
self.mu = mu
self.sigma = sigma
self.rng = np.random.default_rng(seed)
def sample(self, time_vector):
times = np.asarray(time_vector, dtype=float)
if times.ndim != 1 or times.size == 0:
raise ValueError("Provide a nonempty, one-dimensional time vector.")
if (not np.all(np.isfinite(times)) or times[0] < 0
or np.any(np.diff(times) <= 0)):
raise ValueError("Times must be finite, nonnegative, and increasing.")
# Include the first interval from 0, even when sampling starts later.
dt = np.diff(np.concatenate(([0.0], times)))
increments = self.rng.normal(size=times.size) * np.sqrt(dt)
return self.mu * times + self.sigma * np.cumsum(increments)
def sample_path(self, t_0=0.0, t_1=1.0, nofpoints=1000):
if nofpoints < 2 or t_1 <= t_0:
raise ValueError("Use at least two points and t_1 > t_0.")
times = np.linspace(t_0, t_1, nofpoints)
return times, self.sample(times)
motion = BrownianMotion(seed=42)
times, values = motion.sample_path()
This samples the process at finitely many times; joining the points in a plot gives an approximation of a continuous path.
A standard Brownian-motion plot from the original notebook.