The expected number of level crossings for certain symmetric αà-stable processes

The authors give an explicit formula for the first passage time density for some absolutely continuous processes X which can be written as X(t)=Gℝ1(t)+ℝ2(t), where ℝ1, ℝ2 have absolutely continuous sample paths and the random variable G is independent of the processes ℝ1, ℝ2. They use the formula to prove Rice’s formula for the expected number of crossings of a level in a finite interval for certain absolutely continuous symmetric α-stable processes. Numerical examples serve to illustrate the present results.