Calculating quadratic covariance

Question

How can i calculate quadratic covariance $<t^{2}B^{1}_{t},t^{3}B^{2}_{t}>_{T}$ when $B^{1} ,B^{2}$ is independent brownian motion.

Answer 1

You can use the Itô-formula for time dependent functions to get $$t^2B^1_t=\int_0^t2sB^1_s\Bbb ds+\int_0^ts^2\Bbb dB^1_s $$ and analogous for $t^3B^2_t$. As the first integral is a finite variation process you can neglect it in the quadratic covariation and analogous for $t^3B^2_t$. This leads us to $$\langle t^{2}B^{1}_{t},t^{3}B^{2}_{t}\rangle_{T}=\langle \int_0^ts^2\Bbb dB^1_s,\int_0^ts^3\Bbb dB^2_s\rangle_{T}= \int_0^Ts^5\Bbb d\langle B^1,B^2\rangle_{s}. $$ Since the quadratic covariation for two independent Brownian motions is the constant zero function $\langle B^1,B^2\rangle_{s}=0$, the result is zero.