I have recently sat an exam that had elements of stochastic calculus, but I am now feeling like I might have gone wrong in some questions of it like the following.
I am trying to evaluate $\mathbb{E}(B^2_t B^2_s)$, where $B_s$ and $B_t$ are Browninan processes which are not independent. Moreover, all we know is that $t,s>0$.
I have started off by taking an expansion and then simplifying it using the increment rules: $d(B_s^2B_t^2)=2B_sB_t^2dBs + 2B_tB_s^2dBt + B_t^2(dB_s)^2 + B_s^2(dB_t)^2+4B_tB_sdB_sdB_t=\\ =2B_sB_t^2dBs + 2B_tB_s^2dBt + B_t^2ds + B_s^2dt+4B_tB_sdB_sdB_t$
Once this was all done, I have integrated this and finally taken the expectation of the integral, which simplifies by the properties of Itô integrals:
$\mathbb{E}(B^2_t B^2_s)=\mathbb{E}(2\int_0^sB_qB_t^2dB_q)+\mathbb{E}(2\int_0^tB_qB_s^2dB_q)+\mathbb{E}(\int_0^sB_t^2dq)+\mathbb{E}(\int_0^tB_s^2dq)+\mathbb{E}(4\int_0^t(\int_0^sB_qB_rdB_q)dB_r).$
And by the properties of Itô integrals this should simplify to (under Fubini's theorem):
$\mathbb{E}(B^2_t B^2_s)=0+0+\int_0^s\mathbb{E}(B_t^2)dq+\int_0^t\mathbb{E}(B_s^2)dq+0= \int_0^st dq+\int_0^tsdq = 2ts$
Does this make any sense at all? I am not sure if I have assumed anywhere that really they are independent Brownian processes?
I have also solved $\mathbb{E}(B^3_t B^3_s)$, but I might have misunderstood the conditions that allow the expansions of $B_s$ and $B_t$ independently when they are not independent of each other.
Can any kind soul clarify this?