Category Archives: Quadratic Variation

Cross-quadratic variation: correlated Brownian Motions

Let \(W_t\) and \(B_t\) be two independent standard Brownian Motions. For \(\rho \in [0,1]\) define
\[ Z_t = {\rho}\, W_t +\sqrt{1-\rho^2}\, B_t\]

  1. Why is \(Z_t\) a standard Brownian Motion ?
  2. Calculate  the cross-quadratic variations \([ Z,W]_t\) and \([ Z,B]_t\) .
  3. For what values of \(\rho\) is \(W_t\) independent of \(Z_t\) ?
  4. ** Argue that two standard Brownian motions  are independent if and only if  their cross-quadratic variation is zero.