On a stochastic process with switchings

We study a stochastic process $X(t)$ with switchings between two stationary processes with independent increments while achieving regulatory barriers. We obtain the dual Laplace–Stieltjes transform of the distribution of the process $X(t)$ and its limit as $t\to\infty$. Under Cramer's type conditions, the asymptotic representations of these transforms are obtained when the width of the regulating strip is growing. We use known results for regenerative processes and factorization technique for the study in boundary crossing problems for stochastic processes.