= Solution
The <risk function> of an estimator $\widehat\theta$ under the stated <quadratic loss> is
$$
R(\theta,\widehat\theta)
=\mathbb E_\theta
\lVert\widehat\theta(X)-\theta\rVert_2^2.
$$
The <multivariate normal density> is proportional, as a function of $\theta$, to
$$
\exp\left(-\frac12\lVert X-\theta\rVert^2\right),
$$
so <maximum likelihood estimation> gives $\widehat\theta_{\rm MLE}=X$. Writing $X=\theta+Z$ with $Z\sim N_p(0,I_p)$,
$$
\boxed{R(\theta,\widehat\theta_{\rm MLE})
=\mathbb E\lVert Z\rVert^2=p}
$$
for every $\theta\in\mathbb R^p$.
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