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!