For a Brownian motion $B$, set $T_a=\inf\{t\ge 0: B_t=a\}$. Then we can say $B_{t\land T_a}$ is bounded above by $|a|$.
My question is that for Gambler's ruin and martingale
set-up for the gambler's ruin problem:
$(X_n)_{n\geq 1}$ are i.i.d. rv with $P(X_1=1)=1-P(X_1=-1)=p$ and $p\in (0,1),\ p\neq 1/2$. We have integers $0<a<b$, a sequence $S_0:=a$ with $S_n:=S_{n-1}+X_n\quad n\geq 1$ and $\mathcal{F}_n=\sigma(X_1,\dots,X_n)$ and $T$ is the stopping time when either $S_n=0$ or $S_n=b$.
I had to show that the following two are martingales, so this is known: $$M_n:=\left(\frac{1-p}{p}\right)^{S_n}\qquad N_n:=S_n -n(2p-1)$$
Can we say $M_{n\land T}$ and $N_{n\land T}$ bounded above (such as $M_{n\land T}$ bounded above by $\left(\frac{1-p}{p}\right)^{b}\lor 1$ and $N_{n\land T}$ bounded above by $b-T(2p-1)\lor -T(2p-1)$)? But I am not sure about $N_{n\land T}$.
Then they are all uniformly integrable?
The bounds you have written are indeed true (for the second one, there is no need to write the max though), and you can give similar bounds from below: $$0\leq M_{n\wedge T}\leq 1\vee \left(\frac{1-p}{p}\right)^b, $$ and, $$ -T(2p-1)\leq N_{n\wedge T}\leq b-T(2p-1). $$ These are uniform (random) bounds in $n$ that are integrable, since $T$ is. Thus they are all uniform integrable.