I want to show the following:
$$[\int _0 ^t B_{2,s} dB_{1,s}]^2 - \int_0^t (B_{2,s})^2 ds$$ is a martingale where $ (B_{2,s},B_{1,s}) $ is a two dim'l Brownian motion.
My attempt: By Ito formula, $$[\int _0 ^t B_{2,s} dB_{1,s}]^2 =2 \int_0^t (\int_0^u B_{2,s} dB_{1,s} )B_{2,u} dB_{1,u} + \int_0^t (B_{2,u})^2 du$$ so it suffices to show $ \int_0^t (\int_0^u B_{2,s} dB_{1,s} )B_{2,u} dB_{1,u}$ is a martingale. I tried to show the integral of the expectation of the square of the integrand is finite, but it gets me nowhere.
Any help is appreciated.