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How Can i construct a stochastic process $X_t$ which has the property that:

  • $X_t \in [0,1]$ for all $t \in [0,T]$ and
  • $m(\{t \in [0,T] : X_t>0 \})\leq \delta$, for some pre-chosen $\delta \in [0,T]$?

I was thinking of combining Brownian excursions with some stopping times but I'm not sure how to accomplish this..

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  • $\begingroup$ Try conditioning a brownian motion $W_t$ on the event that $W_t$ takes values in $[0,1]$ and on the event that $m({t \in [0,T] : X_t>0})\leq \delta$. $\endgroup$
    – ABIM
    Commented Nov 7, 2016 at 3:35
  • $\begingroup$ How can i get an expression for this in closed form though... $\endgroup$
    – Rombaldo Meniscus
    Commented Nov 7, 2016 at 5:42

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