Is every càdlàg process locally bounded?

Question

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.

Answer 1

The following is an example of càdlàg process, which is not locally bounded.

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.