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Let X be a Levy process with no positive jumps and $\tau_y:=\inf\{t> 0: X_t > y\}$ then we have

$$X_{\tau_y}=y\text{ on }\{\tau_y <\infty\}.$$

Could you explain that why? and does it hold for Levy process with no negative jumps? If X be a Feller process with no positive jumps then does this hold? How about if we state with $T_y:=\inf\{t>0: X_t=y\}.$ Thank you!

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    I don't think it's true unless you assume that $X_0\leq y$. Then, if you consider the process with $X_0\leq y$ and no positive jumps, the idea is that the process can reach $(y,\infty)$ only by the continuous dynamics (I cannot give a formal proof though). If the Levy process does not have negative jumps but can have positive, it is not true: take the Poisson process $N_t$ and put $y = \frac12$. For the case $$ T_y:=\inf\{t>0:X_t = y\} $$ the statement $X_{T_y} = y$ should hold for any process with trajectories continuous from the right.2012-02-17

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