Covariance of Brownian Motion

Assume $s\le t$ and write $B_t=B_s+(B_t-B_s)$.

Use independence of increments to simplify the covariance.

1. 1. Decompose using increments

   For $s\le t$, write $B_t=B_s+(B_t-B_s)$, where the increment $B_t-B_s$ is independent of $B_s$.

2. 2. Compute the covariance

   Since $\text{Var}(B_s)=s$ and the increment contributes zero covariance, the result follows.

   $$\text{Cov}(B_s,B_t)=\text{Cov}(B_s,B_s)+\text{Cov}(B_s,B_t-B_s)=s=\min (s,t)$$

Answer: min(s,t)