The conditional <Jensen inequality> states that for an integrable $X$ and convex $\phi$, whenever the terms are integrable,
$$
\phi(\mathbb E[X\mid\mathcal G])
\leq\mathbb E[\phi(X)\mid\mathcal G]
\quad\text{almost surely}.
$$
For differentiable $\phi$, the supporting-line inequality $\phi(x)\geq\phi(m)+\phi'(m)(x-m)$ with $m=\mathbb E[X\mid\mathcal G]$ gives the result after conditional expectation. Approximation by supporting affine functions proves the general convex case.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-201/3/d/solution