The first passage time of a stable process conditioned to not overshoot

Consider a stable L\'evy process $X=(X_t,t\geq 0)$ and let $T_x$, for $x>0$, denote the first passage time of $X$ above the level $x$. In this work, we give an alternative proof of the absolute continuity of the law of $T_x$ and we obtain a new expression for its density function. Our approach is elementary and provides a new insight into the study of the law of $T_x$. The random variable $T_x^0$, defined as the limit of $T_x$ when the corresponding overshoot tends to $0$, plays an important role in obtaining these results. Moreover, we establish a relation between the random variable $T_x^0$ and the dual process conditioned to die at $0$. This relation allows us to link the expression of the density function of the law of $T_x$ presented in this paper to the already known results on this topic.