# mean variance minimizer

I need to use the lagragian multiplier to find the minimal martingale measure from the set of equivalent martingale measures. i formed the lagragian as L = $L(u(t,S(t)),\lambda) = E_\mathbb{P}[\dfrac{d\mathbb{Q}}{d\mathbb{P}}(H - V(T))^2] + \lambda(\sigma(t,S(t))u(t,S(t)) - \alpha(t,S(t)) + r(t))$ where H is the payoff of an option and $V(T)$ is the terminal value of a self financing portfolio and $\dfrac{d\mathbb{Q}}{d\mathbb{P}} = \exp \{ \int_0^T udB(s) + \frac{1}{2} \int_0^T u^2 ds\}$ $\quad$ I need to slove for the above minimistion problem to get $u$. I have been trying this for more than a week now and i dont seem to yield any reasonable result.Can someone pls help me out with it especially how to compute this $E_\mathbb{P}[\dfrac{d\mathbb{Q}}{d\mathbb{P}}(H - V(T))^2]$ .I think when i get that i can now take the partial derivatives to get my $u$ .Thank you

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