Exercise: Covariance Structure of Brownian Motion

Prerequisites: Brownian Motion, Expectation and Variance

Problem

Let $(W_t)_{t \ge 0}$ be a standard Brownian motion. Prove, using only the four axioms (BM1)–(BM4), that for all $0 \le s \le t$:

Then:

1. Verify the identity numerically by simulation. Take $s = 0.3$, $t = 0.7$, generate $N = 10^5$ sample paths of Brownian motion on $[0, 1]$ using a grid of $\Delta t = 10^{-3}$, and compute the sample covariance of $(W_s, W_t)$. Compare to the theoretical value $\min(s, t) = 0.3$.
2. Using the covariance identity, compute $\operatorname{Corr}(W_s, W_t)$ for $0 < s < t$. Show that the correlation depends only on the ratio $s/t$, not on $s$ and $t$ separately.
3. Use the correlation formula to explain why Brownian motion has long-range dependence in the sense that $\operatorname{Corr}(W_s, W_t)$ does not vanish as the gap $t - s$ grows when both $s$ and $t$ grow proportionally.

Hint

For the proof, write $W_t = W_s + (W_t - W_s)$ and use (BM2) and the mean-zero Gaussian increment from (BM3). For the simulation, use $W_{t_{k+1}} = W_{t_k} + \sqrt{\Delta t}\,Z_k$ with $Z_k \sim \mathcal{N}(0, 1)$ i.i.d.