Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 4 / a / Solution

Write $W=Z-\mathbb{E}Z$ and $\psi (\lambda )=\log {\mathbb{E}}_P{e}^{\lambda W}$. For any $Q\ll P$ and real $\lambda$, define the exponential tiltPositivity of relative entropy givesThus every $Q$ with ${\mathbb{E}}_{Q}W\ge t$ has $D(Q\Vert P)\ge {\sup }_{\lambda \ge 0}\{\lambda t-\psi (\lambda )\}={\psi }^{\ast }(t)$.

For $0<t<b$, continuity of ${\psi }^{\prime }$ supplies ${\lambda }_t>0$ with ${\psi }^{\prime }({\lambda }_t)=t$. Under $P_{{\lambda }_t}$, $\mathbb{E}W=t$, and direct substitution givesThis tilt attains the constrained infimum and proves the identity.