# Is every càdlàg process locally bounded?

Setting We work on a filtered probability space with finite time horizon $$T$$ and let $$X=(X_t)_{t\in[0,T]}$$ be an adapted càdlàg process.

Question Is $$X$$ locally bounded? "Locally bounded" means that there exists a sequence of stopping times $$(T_n)$$ with $$T_n \nearrow T$$ and $$\lim_{n\rightarrow}P[T_n=T]=1$$ and a sequence of constants $$(c_n)$$ such that $$X1_{]]0,T_n]]}$$ is uniformly bounded by $$c_n$$.

Motivation Every càglàd process is locally bounded. I would like to understand if this still holds for càdlàg processes.

Let $$(\Omega, \mathscr{F},\mathsf{P})$$ be a complete probability space and $$\xi,\eta$$ be mutually independent, $$(0,1)$$-uniformly distributed, random variables. Now let $$\begin{equation*} X_t=\xi^{-1}1_{[\hspace{-1pt}[\eta,1 [\hspace{-1pt}[ }(t),\qquad 0\le t<1. \tag{1} \end{equation*}$$ Then $$X=\{X_t, 0\le t<1\}$$ is a càdlàg increasing process with $$X_0=0$$. Suppose $$\mathbf{F}=(\mathscr{F}_t,0\le t<1 )$$ is the natural (complete) filtration of $$X$$(About $$X_t$$ in (1) and its natural filtration please refer to S. W. He et al., Semimartingale Theory and Stochastic Calculus, Sci. Press and CRC(1992). p.160, $$\S$$V.5 and p.172, $$\S$$V.5.70), then $$\begin{equation*} \mathscr{F}_t=\{[t<\eta]\cap \mathscr{N}\}\cup\{[\eta\le t]\cap(\sigma(\xi,\eta)\vee \mathscr{N}) \}). \end{equation*}$$ For any stopping time $$T\in\mathscr{T}(\mathbf{F})$$, we have $$\begin{equation*} \mathsf{P}(T\ge \eta)=0 \quad \text{or} \quad 1. \tag{2} \end{equation*}$$ Hence, from (1) $$\begin{equation*} X_T=\xi^{-1}1_{[T\ge \eta]}=\begin{cases} 0,\quad & \text{if}\quad \mathsf{P}(T\ge \eta)=0,\\ \xi^{-1},\quad&\text{if}\quad \mathsf{P}(T\ge \eta)=1. \end{cases} \end{equation*}$$ Therefore, $$X$$ is not locally bounded.