Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 2

Let ${\tau }_n\uparrow \infty$ be a localizing sequence for the discrete-time local martingale $X$. For $t\ge 1$, the increment of the stopped martingale isand $\{{\tau }_n\ge t\}\in {\mathcal{F}}_{t-1}$. Thus, for any $A\in {\mathcal{F}}_{t-1}$,The assumed integrability gives the dominating random variable $|X_t|+|X_{t-1}|$. The dominated convergence theorem now yields $\mathbb{E}[1_A(X_t-X_{t-1})]=0$, or $\mathbb{E}[X_t\mid {\mathcal{F}}_{t-1}]=X_{t-1}$. Iterating the tower property of conditional expectation proves an integrable discrete-time local martingale is a true martingale. The proof uses the discrete-time increment indicator; the conclusion does not extend to continuous time.

Under the standard local martingale definition, $X_0$ is integrable. For a nonnegative $X$, the Fatou lemma and a localizing sequence giveHence every $X_t$ is integrable, and part (a) applies. Therefore a nonnegative discrete-time local martingale is a true martingale. This is stronger than the continuous-time supermartingale conclusion.

Take an increasing localizing sequence ${\tau }_n$ for $X$ and define the stopping timesThe predictable process condition $K_{u+1}\in {\mathcal{F}}_u$ makes $R_n$ a stopping time. For each fixed finite time range, the finitely many coefficients $K_1,\dots ,K_t$ are finite almost surely; hence $T_n\uparrow \infty$ almost surely.

The stopped martingale transform can be writtenIts coefficient is ${\mathcal{F}}_{s-1}$-measurable and bounded in absolute value by $n$. Each summand is consequently integrable with zero conditional expectation given ${\mathcal{F}}_{s-1}$, since $X^{{\tau }_n}$ is a true martingale. Thus $M^{T_n}$ is a true martingale andThe original PDF fixes $M_0=0$; the TeX extraction omits this equality.

For fixed integers $s\le t$, $A_s=\{\tau =s\}$ belongs to ${\mathcal{F}}_s$ and $X_s=0$ there. The supermartingale property and nonnegativity implyTherefore $X_t=0$ almost surely on $A_s$. Taking the finite union over $s\le t$ gives $X_t=0$ on $\{\tau \le t\}$ almost surely; taking the countable intersection over all integer $t$ makes the assertion simultaneous at every time. Thus zero is absorbing for a nonnegative supermartingale in discrete time.