Exit properties of stochastic processes with stationary independent increments

Abstract:

Let $\{ {X_t},t \geq 0\}$ be a real stochastic process with stationary independent increments. For $x > 0$, define the exit time ${T_x}$ from the interval $( - \infty ,x]$ by ${T_x} = \inf \{ t > 0:{X_t} > x\}$. A reasonably complete solution is given to the problem of deciding precisely when ${P^0}\{ {X_{{T_x}}} = x\} > 0$ and precisely when ${P^0}\{ {X_{{T_x}}} = x\} = 0$. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of ${X_t}$. A few applications of this result are discussed.