# Optional stopping theorem: different versions

$$(X_k)_{k \in \mathbb{N}}$$ 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 $$E[X_{R_2}|\mathcal{F}_{R_1}] \geq X_{R_1}$$ ?

• Could you please include full references to the books/papers you are quoting from? Jun 7, 2021 at 15:08
• References added Jun 7, 2021 at 20:34
• @Kurt.W.X, Does still interesting in this question right now? Mar 10, 2022 at 8:00
• Sure, it would be great. Mar 11, 2022 at 5:37

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

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 $$R_2(\ge)R_1$$.

1. 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.

1. 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).