On the scaling property in fluctuation theory for stable L\'evy processes

We find an expression for the joint Laplace transform of the law of $(T_{[x,+\mathrm{\infty }[},X_{T_{[x,+\mathrm{\infty }[}})$ for a L'evy process $X$, where $T_{[x,+\mathrm{\infty }[}$ is the first hitting time of $[x,+\mathrm{\infty }[$ by $X$. When $X$ is an $\alpha$-stable L'evy process, with $1<\alpha <2$, we show how to recover from this formula the law of $X_{T_{[x,+\mathrm{\infty }[}}$; this result was already obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when $X$ is non spectrally negative. Then, we study the behaviour of the time of first passage $T_{[x,+\mathrm{\infty }[},$ conditioned to ${X_{T_{[x,+\infty[}} -x \leq h}$when$h$tendsto$0$\mathrm{.}Thisstudybringsforwardanasymptoticvariable$T_x^0$, which seems to be related to the absolute continuity of the law of the supremum of $X$.