Stopping times and martingales expectation

Question

Suppose $M_n$ is a martingale and $T$ a stopping time. It is well known that $E[M_0]=E[M_n]$ for all $n\geq 0$ but under what assumptions on $T$ does it hold that $E[M_0]=E[M_{T\wedge n}]$? I don't think it holds for a $T$ that can be infinite on a set of positive probability, but I can't prove it, nor tell if it's actually true.

Answer 1

For any martingale $\{M\}_{n\in\mathbb{N}}$ and any stopping time $T$ it is true that $\mathbb{E}[M_0]=\mathbb{E}[M_{T\wedge n}]$. Therefore, we also have $\mathbb{E}[M_0]=\lim_{n\rightarrow \infty}\mathbb{E}[M_{T\wedge n}]$. On the other hand, it is NOT always the case that $\lim_{n\rightarrow\infty}\mathbb{E}[M_{T\wedge n}]=\mathbb{E}[M_T]$. That requires justifying interchanging the order of limits and expectations.

Conditions under which this is true are established by optional sampling theorem. For example, we have $\lim_{n\rightarrow\infty}\mathbb{E}[M_{T\wedge n}]=\mathbb{E}[M_T]$ if any of the following holds:

1. $T$ is bounded, or
2. $\{M\}_{n\in\mathbb{N}}$ is bounded (i.e., there exists $C\in\mathbb{R_+}$ such that for all $n\in\mathbb{N}$ we have $|M_n|\leq C$), or
3. $\{M\}_{n\in\mathbb{N}}$ is uniformly integrable.