I'm given two eqautions $$dS(t)=\mu S(t)dt+\sigma S(t)dW_t$$ $$ dX_t = tdS_t$$ I need to calculate $Var(X_t)$.
My idea is to use the solution of $dS(t)=\mu S(t)dt+\sigma S(t)dW_t$ i.e $S_t$ and Ito's isometry to obtain $$Var(X_t) = ...= \sigma^2 S_0^2 \int_0^t u^2e^{2\mu u + \sigma^2 u^2}du$$ The last integral can be calculated but it's rather complicated. I reackon that the problem can be solved easier.