ON THE PROBABILITY OF A RANDOM PROCESS TO EXIT FROM THE INTERVAL

Abstract

This article studies the probabilities of the first exit from a given interval for a stochastic process consisting of a positive-deviation Wiener process and a generalized Poisson process with exponentially distributed negative jumps. The model under consideration is important in describing complex random systems encountered in insurance mathematics, risk theory, financial mathematics, and queuing theory. In this work, using the factorization method, precise analytical formulas are obtained for the probabilities of loss corresponding to the exit of the process from the interval boundaries. The properties of the Laplace exponent of the process are studied, and closed-form expressions of the probabilities are derived using the real roots of the characteristic equation. The results obtained serve to develop the theory of boundary functionals for stochastic models consisting of a combination of Wiener and jump processes.