The quadratic covariation of Brownian motion and poisson process

[knightzero]

Hi:
I want to know the quadratic covariation of Brownian motion B(t) and poisson process N(t).Is it B(t)?
Thanks !

 

[wayneckm]

I think it should be zero because Brownian motion is a continuous process without jump components while Poisson is a quadratic pure jump process with continuous part being constant. Hence, $$[B,N] = [B,N]^{c} + \Delta B \Delta N = 0 + 0 = 0$$ where $$[B,N]^{c}$$ is the continuous part of quadratic covariation process.