Covariance of Two Brownian Times

Stochastic Processes

Brownian Motion

Easy

For standard Brownian motion, find

Cov (B_{2}, B_{7})

Hint 1

Write $B_{7} = B_{2} + (B_{7} - B_{2})$ and use independence of the increment from $B_{2}$ .

Hint 2

The general formula is $Cov (B_{s}, B_{t}) = min (s, t)$ .

Show worked solution

1. Decompose the later time
Since $B_{7} = B_{2} + (B_{7} - B_{2})$ and the increment $B_{7} - B_{2}$ is independent of $B_{2}$ , $Cov (B_{2}, B_{7}) = Var (B_{2})$ .
2. Evaluate
Since $Var (B_{2}) = 2$ , the covariance equals the smaller of the two times.
$Cov (B_{2}, B_{7}) = min (2, 7) = 2$

Answer: 2

1. Decompose the later time