The First Passage Time of a Stable Process Conditioned to Not Overshoot

Consider a stable Lévy process $$X=(X_t,t\ge 0)$$ X = ( X t , t ≥ 0 ) and let $$T_{x}$$ T x , for $$x>0$$ x > 0 , denote the first passage time of $$X$$ X above the level $$x$$ x . In this work, we give an alternative proof of the absolute continuity of the law of $$T_{x}$$ T x and we obtain a new expression for its density function. Our constructive approach provides a new insight into the study of the law of $$T_{x}$$ T x . The random variable $$T_{x}^{0}$$ T x 0 , defined as the limit of $$T_{x}$$ T x when the corresponding overshoot tends to $$0$$ 0 , plays an important role in obtaining these results. Moreover, we establish a relation between the random variable $$T_{x}^{0}$$ T x 0 and the dual process conditioned to die at $$0$$ 0 . This relation allows us to link the expression of the density function of the law of $$T_{x}$$ T x presented in this paper to the already known results on this topic.