Brownian Bridge Covariance

Worked Example

Consider a standard Brownian motion process conditional on starting at W(0) = a and ending at W(T) = b. The noise scale is σ. Let's calculate the conditional mean, variance, and covariance for a specific scenario.

Assume a = -1, b = 1 ("Ascending"), T = 2, q = t/T = 0.5, u = s/T = 0.25, and σ = 1.

1. Conditional Mean at t = 0.5 · 2 = 1:
   Mean = a + (b - a) · q = -1 + (1 - (-1)) · 0.5 = -1 + 1 = 0.
2. Variance at t:
   Variance = σ² · t · (1 - t/T) = 1² · 1 · (1 - 0.5) = 0.5.
3. Covariance between s = 0.5 and t = 1:
   Covariance = σ² · (min(s, t) - s · t / T) = 1² · (0.5 - 0.5 · 1 / 2) = 0.5 - 0.25 = 0.25.
4. Correlation:
   First, find variance at s: Variance = 1² · 0.5 · (1 - 0.25) = 0.375.
   Correlation = Covariance / √(Var(s) · Var(t)) = 0.25 / √(0.375 · 0.5) = 0.25 / √(0.1875) ≈ 0.57735.

Expected result: Covariance is 0.25.

Experiment

Explore this process in the Brownian Bridge Covariance module.

1. Set the Endpoints to "Ascending", Duration to 2.
2. Set Time fraction to 0.5 and Other time fraction to 0.25.
3. Observe the Covariance is exactly 0.25 and Correlation matches the calculation above.
4. Change the Time fraction to 0. This moves the evaluation point exactly to the starting endpoint.
5. Observe the Variance becomes 0 because the starting endpoint is pinned. The Correlation becomes "Unavailable" as division by zero variance is not defined.

Limitations

This visualization shows a discrete number of sample paths generated using a pseudo-random number generator (mulberry32) and the Box-Muller transform for normal variates. In reality, Brownian motion is a continuous limit.

Furthermore, the sample paths shown are drawn using finite time steps. Between these simulated steps, the mathematical Brownian bridge fluctuates infinitely often (non-differentiability), which is not fully captured by connecting discrete points with straight lines on a pixel grid.

When the noise scale σ is 0, or at the exact endpoints, the variance drops to identically zero. Correlation becomes mathematically undefined in these regions.

References

For a rigorous treatment of Brownian bridge covariance structures and conditioning, refer to:

• MIT 18.175 Lecture 29 Notes, which formalizes the properties of Brownian diffusions conditioned on their endpoints.