Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 3 / d / Solution

The conditional Jensen inequality states that for an integrable $X$ and convex $\varphi$, whenever the terms are integrable,For differentiable $\varphi$, the supporting-line inequality $\varphi (x)\ge \varphi (m)+{\varphi }^{\prime }(m)(x-m)$ with $m=\mathbb{E}[X\mid \mathcal{G}]$ gives the result after conditional expectation. Approximation by supporting affine functions proves the general convex case.