Can't understand the order of computing the next integral where B is real Brownian motion with $0\leq s \leq t$ by using Fubini-Tonelli: $$cov(X_s,X_t)=E\biggl( \int_0^sB_rdr\int_0^tB_udu \biggr)=E\biggl(\biggl( \int_0^sB_rdr+\int_s^tB_rdr \biggr)\cdot\int_0^sB_udu\biggr)=I_1+I_2$$ I compute two integrals separately: $$ I_2=E\biggl( \int_s^t\int_0^sB_rdrB_udu \biggr)= \int_s^t\int_0^sE( B_rB_u)drdu =\int_s^t\int_0^s(min(r,u))drdu=\int_s^t\int_0^sududr $$ $$I_1=E\biggl( \int_0^s\int_0^sB_rdrB_udu \biggr)=E\biggl( \int_0^s\biggl(\int_0^r B_rB_udu \biggr)dr\biggr)+E\biggl( \int_0^s\biggl(\int_r^s B_rB_udu \biggr)dr\biggr)=$$ $$=J_1+J_2$$ and the same integration as previously for $J_1$ and $J_2$.
Here are my questions:
- why should I split $I_1$ again into two integrals? But not compute expectation as taking minimum among $B_r$ and $B_u$ straight away as integration is between $0$ and $s$;
- then why shouldn't I split the $J_2$ again? But in this case allowed to compute expectation by taking minimum;
- and the last one why the minimum in $J_1$ is $u$ and in $J_2$ is $r$?