Hi,

Having a random variable $X$ I am trying to find a stochastic process $Z_t$ such that:

$$P[Z_t>T] = P[X > T | X > t]$$

for all $T>t$, or a proof that such a process does not exist.

Please note that this question is not related to any homework and that I actually need this result for my research in financial maths.

Edit I haven't really mentioned it, but what I am really after is some sort of closed form formula for $Z_t$, ideally as a function of $X$ and $t$.