- \mu_n|^p \| \F_n) \leq 
   2^p \E (|\DE_n|^p + |\mu_n|^p \| \F_n) \leq 2^p (B + (1 + B)^p) .$$ Applying Theorem \[th:2\] to $\{ M_n \}$ with $b = 2^p (B + (1+B)^p)$ and $M_0 := Y_0$ yields $$\label{eq:C}
\E \left( (M_t^+)^r \one_{\tau > t} \right) \leq C t^{r-p} .$$ When $\sigma > t$ it follows that $M_n \geq n + A_n$ for $1 \leq n \leq t$ and hence that $\tau > t$. Also, when $\sigma > t$, we know that $M_t = Y_t + A_t \geq Y_t$ and therefore that $$Y_t^+ \one_{\sigma > t} \leq M_t^+ \one_{\tau > t} .$$ The conclusion of the corollary now follows from (\[eq:C\]), with $K = C (2^p (B + (1+B)^p) , p , r)$. $\Cox$

The above argument uses no properties of the process $A_n$, other than its being nonincreasing and adapted. In particular, it need not be predictable. If the increments $\DE_n$ can be decomposed into the sum of two parts, one satisfying the hypotheses of the corollary and one nonincreasing and adapted, then the second of these can be absorbed into the process $\{ A_n \}$ and the result will still hold. Without loss of generality, the second 