Show that a local martingale is a true martingale if and only if it is a process of class DL

Question

Let $M$ be a local martingale. Show that $M$ is a (true) martingale if and only if it is a process of class DL.

Quick definitions: $\mathscr{S}_a$ is the class of all stopping times $T$ such that $P(T \le a) = 1$. DL is the class of all right-continuous processes $\{X_t\}_t$ such that $\{X_T\}_{T \in \mathscr{S}_a}$ is uniformly integrable for every $0 < a < \infty$.

Since $M$ is a local martingale, there exists a sequence of stopping times $\{S_n\}$ such that $S_n \uparrow \infty$ and $M_t^{(n)} \triangleq M_{t \land S_n}$ is a martingale.

We can define a variant $S_n' = S_n \land n$ such that $S_n'$ is in $\mathscr{S}_n$ and that $S_n'$ is still nondecreasing and $S_n' \uparrow \infty$. Then $M_t^{(n)'} \triangleq M_{t \land S_n'}$ is a martingale.

If $M$ is of class DL, then for every $n \ge 0$, the family $\{ M_{T} \}_{T \in \mathscr{S}_n}$ is uniformly integrable.

I'm stuck on where to go from here.

Answer 1

1. Suppose that the $M$ is a local martingale with respect to the filtration $\{ {\mathcal F}_s\}$ and the stopping times $S_n \uparrow \infty$. Then for $s<t$, we have $$\forall n, \quad E[M_{t \wedge S_n} \mid {\mathcal F}_s]=M_{s \wedge S_n} \quad (*) \,. $$ Now assume that $M$ is also DL. Then the variables $\{M_{t \wedge S_n}\}_{n \ge 1}$ are uniformly integrable and converge a.s. to $M_t$, so they also converge in $L^1$ to $M_t$. Jensen's inequality implies that conditional expectation is a contraction on $L^1$ (See [1] or [2]) so taking $n \to \infty$ in (*) gives $$ E[M_{t } \mid {\mathcal F}_s]=M_{s } \,, $$ so $M$ is a Martingale.

2. Conversely, suppose $M$ is a Martingale with respect to the filtration $\{ {\mathcal F}_s\}$ in $(\Omega, {\mathcal F},P)$. Then for $a>0$ and each stopping time $T \le a$, the definition of conditional expectation implies that $$ E[M_a \mid {\mathcal F}_T]=M_T \quad (**)\,,$$ where $${\mathcal F}_T:= \{A \in {\mathcal F}:\, \forall t \ge 0, \; A \cap \{T \le t\} \in {\mathcal F}_t \}\,. $$ Thus for every $0 < a < \infty$, all the variables $\{M_T\}_{T \in \mathscr{S}_a}$ are conditional expectations of $M_a$ under different $\sigma$-fields, hence they form a uniformly integrable family, see [2] or [3] or [4]. Thus $M$ is DL.

[1] https://www.math.purdue.edu/~stindel/teaching/ma539/cdt-expectation-2.pdf page 41 [2] Probability: Theory and examples by R. Durrett [3] uniform integrability of all conditional expectations of a fixed $L^1$ function [4] https://planetmath.org/conditionalexpectationsareuniformlyintegrable