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Let $A=V^{1/2}$ be the symmetric positive square root of the covariance matrix, and write $S_1=\mu +AZ$ with $Z$ having the multivariate normal distribution $N_d(0,I)$. Bounded gradient makes $g$ globally Lipschitz, so $g(S_1)$ has at most linear growth and is square integrable. This also justifies the permitted Gaussian integration by parts formula for each coordinate.

By the chain rule, ${\nabla }_{z}g(\mu +Az)=A^T\nabla g(\mu +Az)$. Applying Gaussian integration by parts givesThus the Gaussian quadratic hedge holds ${\pi }^{\ast }=\mathbb{E}[\nabla g(S_1)]$. Substitution into its initial capital formula yieldsOnly the hedge calculation is being used here; the Gaussian model need not possess a positive pricing weight for this least-squares identity.