Martingale Stopping Time

Question

Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | \mathcal F_n) < c$.

How do I prove that $(S_{\tau \wedge n})_{n \ge 0}$ is a uniformly bounded martingale, and that $\mathbb E(S_\tau) = \mathbb E (S_0)$?

Answer 1

Hint

$$S_{\tau \wedge n} = S_0 + \sum_{j=1}^{(\tau \wedge n)} (S_{j}-S_{j-1}) = S_0 + \sum_{j=1}^n (S_j-S_{j-1}) \cdot 1_{\{\tau>j-1\}}$$

implies

$$\mathbb{E}(|S_{n \wedge \tau}|) \leq \mathbb{E}(|S_0|)+c \cdot \mathbb{E}(\tau)<\infty $$

To prove $\mathbb{E}(S_{\tau}) = \mathbb{E}(S_0)$, use optional stopping and dominated convergence (which works fine because of the uniform bound).

Answer 2

Reference: See Theorem 5.7.5 (p. 230) of Probability: Theory and Examples (4th edition) by Richard Durrett. The book is freely available at the author's website.