V
\eqno(\mbox{\rm C2})$$ Then for any $r \in (0 , p-1)$ there is a $c = c (p , a, V, J , r) > 0$ such that $\E (X_n^+)^r < c$ for all $n$.

Applying this theorem to the process $X_n' := X_n - (X_0-J)^+$ in the case $r = 1$ immediately yields the corollary:

\[cor:X\_0\] Under hypotheses (C1) and (C2) of Theorem \[th:1\], but without assuming $X_0 \le J$, we have $$\E (X_n \| \F_0) \leq c(p,a,V,J,1) + (X_0-J)^+ .$$

**Remark. By following through the proof (presented in Sections 4 and 5), we are able to provide an explicit formula for the quantity $c(p,a,V,J,r)$ of Theorem 1. Indeed, for $a=1$ and $J=0$, we have $c(p,1,V,0,r) = K \zeta(p-r)$ where $K = C(b,p,r) = 2^{p/2} c_2 C'(p,b) + c_4$. Here $b = 2^p (B + (1+B)^p)$; $B = 2^p (1+V)$; $C'(p,b) = \max(1, \ c'(p,b))$; $c'(p,b) = c_p b (1+c_p^{-1})^p$; $c_2 = c_p b (4^p + 4^{p-r} {r \over p-r})$; $c_4 = C'(p,b) c_3 \zeta(p/2)$; $c_3 = 3^r 4^p b (c_p b + {p \over p-r} + 3^r)$; and $c_p = (p-1)^p$ is the constant from Burkholder’s inequality. (Recall that $\zeta(w) \equiv \sum_{i=1}^\infty i^{-w}$ is the Riemann zeta function, 