A discontinuous Galerkin method for approximating the stationary distribution of stochastic fluid-fluid processes

Abstract

Introduced by Bean and O'Reilly (2014), a stochastic fluid-fluid process is a Markov processes $\{X_t,Y_t,{\phi }_t{\}}_{t\ge 0}$, where the first fluid $X_t$ is driven by the Markov chain ${\phi }_t$, and the second fluid $Y_t$ is driven by ${\phi }_t$ as well as by $X_t$. That paper derived a closed-form expression for the joint stationary distribution, given in terms of operators acting on measures, which does not lend itself easily to numerical computations. Here, we construct a discontinuous Galerkin method for approximating this stationary distribution, and illustrate the methodology using an on-off bandwidth sharing system, which is a special case of a stochastic fluid-fluid process.