computation is to bound the quantity $\E ((M_t^+)^r \one_{T = k , M_t > t} \| \F_k)$. Break this into three pieces: the part up to time $k$, the jump at time $k$, and the part from time $k+1$ to time $t$. For any $0 < r < p-1$, $|x+y+z|^r \leq 3^r (|x|^r + |y|^r + |z|^r)$ (use convexity when $r \geq 1$ and sublinearity when $r \leq 1$). Hence $$(M_t^+)^r \leq 3^r \left[ (M_k^+)^r + (\DD_k^+)^r + ((M_t - M_{k+1})^+)^r 
\right] .$$ The event $\{ T = k \}$ implies $M_k \leq 3t/4$, and is also in the initital $\sigma$-field of the martingale $\{ M_n - M_{k+1} : 
n \geq k+1 \}$. Therefore, when we condition on $\F_k$, we get $$\begin{aligned}
\E ((M_t^+)^r \one_{T = k , M_t > t} \| \F_k) & \leq & 3^r \left [ 
   \left ( {3t \over 4} \right )^r \P (\DD_k \geq t/4 \| \F_k) \right. \\
&& + \E ((\DD_k^+)^r \one_{\DD_k \geq t/4} \| \F_k) \\
&& + \left. \E (|M_t - M_{k+1}|^r \one_{T = k , M_t - M_{k+1} \geq t/4} 
   \| \F_k) \right ] \end{aligned}$$

The moment condition $\E (|\DD_k|^p \| \F_k) \leq b$ implies that $\P (\DD_k \geq y) 