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

The stochastic fluid-fluid model (SFFM) is a Markov process {(X_t, Y_t, φ_t), t ≥ 0}, where {φt, t ≥ 0} is a continuous-time Markov chain, the first fluid, {X_t, t ≥ 0}, is a classical stochastic fluid process driven by {φ_t, t ≥ 0}, and the second fluid, {Y_t, t ≥ 0}, is driven by the pair {(X_t, φ_t), t ≥ 0}. Operator-analytic expressions for the stationary distribution of the SFFM, in terms of the infinitesimal generator of the process {(X_t, φ_t), t ≥ 0}, are known. However, these operator-analytic expressions do not lend themselves to direct computation. In this paper the discontinuous Galerkin (DG) method is used to construct approximations to these operators, in the form of finite dimensional matrices, to enable computation. The DG approximations are used to construct approximations to the stationary distribution of the SFFM, and results are verified by simulation. The numerics demonstrate that the DG scheme can have a superior rate of convergence compared to other methods.