Observability of continuity for a RCLL process

Question

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $X:[0;+\infty)\times\Omega\to\mathbb{R}^d$ be a stochastic process on this probability space with the property that all its sample paths are RCLL. Let $t_0>0$. Let $A=\{\omega\in\Omega | t\mapsto X_t(\omega) \text{ is a continuous function on } [0;t_0)\}$. Is it possible to prove that $A\in\mathcal{F}^X_{t_0}$, where $\left(\mathcal{F}^X_t\right)_{t\geq 0}$ is the filtration generated by process $X$, i.e. $\mathcal{F}^X_t=\sigma(X_s;0\leq s \leq t)$ ?

This is a question I encountered in Karatzas & Shreve (Exercise 1.1.7.).

Answer 1

Without loss of generality, $d=1$; if not examine the coordinates of $X$ separately, or consider the rcll $(\mathcal F^X_t)$-adapted process $t\mapsto|X_t|$.

Consider the oscillation $Y_t(\omega,\delta):=\sup_{s:|t-s|\le\delta}X_s(\omega)-\inf_{s:|t-s|\le\delta}X_s(\omega)$. Because $X$ is rcll, the supremum and infimum here can be taken over rational $s$. Consequently, $Y_t(\omega,\delta)$ is $\mathcal F_{t+\delta}^X$-measurable.

Fix $t>0$ and define $$ \eqalign{ B(t):&=\{\omega: X_{s-}(\omega)\not=X_s(\omega)\hbox{ for some }s\in[0,t]\}\cr &=\cup_{m=1}^\infty\cap_{n=1}^\infty\cup_{k=1}^n\{\omega: Y_{t(k-1)/n}(\omega,1/n)>1/m\}.\cr } $$ (In effect, if $X$ has a jump in the time interval $[0,t]$ of size exceeding $1/m$, then for each $n$ the oscillation of $X$ over $[t(k-1)/n,t/n]$ must exceed $1/m$ for some $k\in\{1,2,\ldots,n\}$.) Notice that $B(t)$ is $\mathcal F^X_{t+}$-measurable. Finally, $$ A=\cup_{q<t_0,q\in\Bbb Q}B(q)\in \mathcal F^X_{t_0}. $$