Stochastic integration of $s^2$ with respect to the Brownian motion

Question

When applying integration by part formula for Itô's integrals, I get $$\int_0^T s^2dB_s=T^2B_T-\int_0^T2sB_sds$$ but how do we deal with the term $\int 2sB_sds$?

Answer 1

I wouldn't try to deal with the $\int_0^T 2s B_s ds$ term directly. Instead notice that $\int_0^T s^2 dB_s$ is the Itô integral of a deterministic $L^2$ function; hence, it's a mean-zero Gaussian process with covariance: $$\mathbb{E}\left[ \int_0^t r^2 dB_r \int_0^s r^2dB_r \right] = \int_0^{\min (t,s)} r^4 dr = \frac{\min(t,s)^5}{5}$$