A Discontinuous Galerkin Method for Approximating the Stationary Distribution of Stochastic Fluid-Fluid Processes

Nigel Bean, Angus Lewis (), Giang T. Nguyen, Małgorzata M. O’Reilly and Vikram Sunkara
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Nigel Bean: The University of Adelaide
Angus Lewis: The University of Adelaide
Giang T. Nguyen: The University of Adelaide
Małgorzata M. O’Reilly: The University of Tasmania
Vikram Sunkara: Freie Universität Berlin

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 4, 2823-2864

Abstract: Abstract The stochastic fluid-fluid model (SFFM) is a Markov process $$\{(X_t,Y_t,\varphi _t),t\ge 0\}$$ { ( X t , Y t , φ t ) , t ≥ 0 } , where $$\{\varphi _t,{t\ge 0}\}$$ { φ t , t ≥ 0 } is a continuous-time Markov chain, the first fluid, $$\{X_t,t\ge 0\}$$ { X t , t ≥ 0 } , is a classical stochastic fluid process driven by $$\{\varphi _t,t\ge 0\}$$ { φ t , t ≥ 0 } , and the second fluid, $$\{Y_t,t\ge 0\}$$ { Y t , t ≥ 0 } , is driven by the pair $$\{(X_t,\varphi _t),t\ge 0\}$$ { ( 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 $$\{(X_t,\varphi _t),t\ge 0\}$$ { ( 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.

Keywords: Stochastic fluid-fluid processes; Stationary distribution; Discontinuous Galerkin method (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-022-09945-2

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