Hitting probabilities and hitting times for stochastic fluid flows

Abstract

Recently there has been considerable interest in Markovian stochastic fluid flow models. A number of authors have used different methods to calculate quantities of interest. In this paper, we consider a fluid flow model, formulated so that time is preserved, and derive expressions for return probabilities to the initial level, the Laplace–Stieltjes transforms (for arguments with nonnegative real part only) and moments of the time taken to return to the initial level, excursion probabilities to high/low levels, and the Laplace–Stieltjes transforms of sojourn times in specified sets. An important feature of our results is their physical interpretation within the stochastic fluid flow environment, which is given. This allows for further implementation of our expressions in the calculation of other quantities of interest. Novel aspects of our treatment include the calculation of probability densities with respect to level and an argument under which we condition on the infimum of the levels at which a “down–up period” occurs. Significantly, these results are achieved with techniques applied directly within the fluid flow model, without the need for discretization or transformation to other equivalent models.