Exercise: Variance Is a Submartingale — Doob's Decomposition

Prerequisites: Martingales (Discrete Time), Conditional Expectation

Problem

Let $(M_n)$ be a square-integrable martingale ($\mathbb{E}[M_n^2] < \infty$) with $M_0 = 0$.

1. Using conditional Jensen's inequality, show that $(M_n^2)$ is a submartingale. Identify precisely where the martingale property gets used versus the convexity of $x \mapsto x^2$.

2. Doob's decomposition. Any submartingale $(X_n)$ can be uniquely written as $X_n = Y_n + A_n$, where $Y_n$ is a martingale ($Y_0 = X_0$) and $A_n$ is a predictable, non-decreasing process with $A_0 = 0$, defined by $A_n = \sum_{k=1}^n \mathbb{E}[X_k - X_{k-1} \mid \mathcal{F}_{k-1}].$ Verify this construction for $X_n = M_n^2$: compute $A_n$ explicitly in terms of the conditional variance $\operatorname{Var}(M_n - M_{n-1} \mid \mathcal{F}_{n-1})$.
3. The process $A_n$ in part 2 is called the predictable quadratic variation (or angle bracket) of $M$, denoted $\langle M \rangle_n$. Show that $\langle M\rangle_n = \sum_{k=1}^n \mathbb{E}[(M_k - M_{k-1})^2 \mid \mathcal{F}_{k-1}]$.
4. Concrete case. For the symmetric random walk $S_n$ with i.i.d. $\pm 1$ increments, compute $\langle S\rangle_n$. What is the submartingale $S_n^2$ and its martingale part $Y_n = S_n^2 - \langle S\rangle_n$?

Hint

Conditional Jensen's inequality: for a convex function $\phi$, $\phi(\mathbb{E}[X \mid \mathcal{G}]) \le \mathbb{E}[\phi(X) \mid \mathcal{G}]$.