A Numerical Truncation Approximation with A Posteriori Error Bounds for the Solution of Poisson’s Equation

Saied Mahdian, Peter W. Glynn and Yuanyuan Liu ()
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Saied Mahdian: Stanford University
Peter W. Glynn: Stanford University
Yuanyuan Liu: Central South University

Methodology and Computing in Applied Probability, 2025, vol. 27, issue 2, 1-18

Abstract: Abstract The solution to Poisson’s equation arises in many Markov chain and Markov jump process settings, including that of the central limit theorem, value functions for average reward Markov decision processes, and within the gradient formula for equilibrium Markovian rewards. In this paper, we consider the problem of numerically computing the solution to Poisson’s equation when the state space is infinite or very large. In such settings, the state space must be truncated in order to make the problem computationally tractable. In this paper, we provide the first truncation approximation solution to Poisson’s equation that comes with provable and computable a posteriori error bounds. Our theory applies to both discrete-time chains and continuous-time jump processes. Through numerical experiments, we show our method can provide highly accurate solutions and tight bounds.

Keywords: Markov chain; Markov jump process; Poisson’s equation; Lyapunov functions; Numerical bounds; State space truncation; 60J10; 60J22; 60J27; 60J74; 65C40 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11009-025-10158-6

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