I'm trying to define a stochastic process $X_t$ with values on $\mathbb{R}$ which has the following properties, for the time interval $t \in [0,T]$:
- $X_t$ takes values in $[0,1]$
- $m(\{t \in [0,T] : X_t>0\})\leq \epsilon$, for some prescribed $\epsilon\geq 0$, where $m$ is the Lebesgue measure on $\mathbb{R}$.
- $X_t$ is adapted to some other Lévy process $Z_t$ which takes values in $\mathbb{R}^d$.
Most importantly I am looking for the following Stopping condition to hold: - Let $\tau_1,...$ and $r_1,...$ be the hitting times corresponding to when $X_t>0$ and when $X_t$ first becomes $0$ again. I want $r_i -\tau_i$ to be small, always but stochastic.
I am not sure how to rigorously construct such a process... Could I just have a sequence of stopping times (this seems to make sense) but then how do I enforce the 2 and 4th conditions?