Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

We derive several algorithms, including quadratically convergent algorithms, which can be used to calculate the Laplace–Stieltjes transforms of the time taken to return to the initial level in the Markovian stochastic fluid flow model. We give physical interpretations of the algorithms and consider their numerical analysis. The numerical performance of the algorithms, which depends on the physical properties of the process, is discussed and illustrated with simple examples. Besides the powerful algorithms, this paper contributes interesting theoretical results. In particular, the methodology for constructing these algorithms is a valuable contribution to the theory of fluid flow models. Moreover, useful physical interpretations of the algorithms, and related expressions, given in terms of the fluid flow model, can assist in further analysis and help in a better understanding of the model. The authors would like to thank the Australian Research Council for funding this research through Discovery Project DP0770388.