Optional stopping theorem: different versions

Question

$(X_k)_{k \in \mathbb{N}}$ is a sub-martingale, $R_1,R_2$ two stopping times such that $R_1 \leq R_2$.

The optional stopping theorem which I know is the following (from probability theory, independence, interchangeability, martingales: theorem 5 page 248, other reference Shiryaev probability 2: theorem 1 page 119):

Theorem $1$ : letting $X_{R_1}(w)=X_{R_1(w)}(w)1_{\{R_1<\infty\}}(w),$ if $X_{R_1},X_{R_2} \in L^1,$ $\liminf_k\int_{\{R_2>k\}}|X_k|dP=0$ then $E[X_{R_2}|\mathcal{F}_{R_1}] \geq X_{R_1}.$

Consider the following theorem from Shreve-Karatzas book:

In the proof of theorem 3.22, he refers to theorem $9.3.5$ from a course in probability theory:

He refers to exercise $11$, which can be proved easily using theorem $1,$ as a counterexample to theorem $9.3.5$ (not true for all cases):

What's the difference between theorem $1$ and theorem $9.3.5$ ? In theorem $3.22,$ is it necessary to define $X_{\infty}$ as the almost sure limit ? What are the weakest conditions to obtain $E[X_{R_2}|\mathcal{F}_{R_1}] \geq X_{R_1}$ ?

Answer 1

I am charmed to see people are still using Chung's book. By the the index that Chung uses, $N_{\infty}$ , he means that the supermartingale is a supermartingale up to and in including $\infty$. If you were to say this about martingales you would mean that there was a limit, $X_{\infty}$, and the $E(X_{\infty}|F_n ) = X_n$, closeable may be the term. This is strong, the martingale is uniformly integrable, and the conditions of your first theorem are guaranteed.$$$$ A counterexample to keep in mind is recurrent random walk, where you might take $\beta = inf \lbrace n>5: S_n = 0 \rbrace $, in which case $S_5, S_{\beta}$ is clearly not a martingale

Answer 2

Suppose that $X=\{X_n,\mathscr{F}_n; n\in N_+ \}$ is a sub-martingale, $R_1,R_2$ are two stopping times such that $R_1\le R_2$. Denote \begin{equation*} \mathsf{E}[X_{R_2}|\mathscr{F}_{R_1}]\ge X_{R_1}. \tag{1} \end{equation*}

1. What's the difference between theorem 1 and theorem 9.3.5:

Both theorem 1 and theorm 9.3.5 give the conditions of (1) holds, but theorem 1 is for the a fixed $R_2$, theorem 9.3.5 give the sufficient conditions of (1) holds for arbitrary stopping time $R_2(\ge)R_1$.

2. Is it necessary to define $X_\infty$ as the almost sure limit?

To get the conclusion of therem 9.3.5, usually, suppose that the (sub/super-)martingale $X$ is right closed(cf. C. Dellacherie & P. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, 1982, p.5--, V.7--V.10). If $X$ is right closed, then \begin{equation*} \lim_{n\to\infty}X_n=X_\infty, \quad X_\infty(\ge/\le)=\mathsf{E}[Y|\mathscr{F}_\infty] \qquad \text{a.s.} \end{equation*} and $X_\infty$ is the $\mathscr{F}_\infty$-measurable right closing element of $X$ also.

3. What are the weakest conditions to obtain (1):

For fixed $R_2$, above mentioned Theorem 1 is the weakest conditions to obtain (1). For arbitrary $R_2$, ``$X$ is right closed'' is the weakest conditions to obtain (1).