High-Order Steady-State Diffusion Approximations

Much like higher-order Taylor expansions allow one to approximate functions to a higher degree of accuracy, we demonstrate that, by accounting for higher-order terms in the Taylor expansion of a Markov process generator, one can derive novel diffusion approximations that achieve a higher degree of accuracy compared with the classical ones used in the literature over the last 50 years. We derive and analyze new diffusion approximations of stationary distributions of Markov chains that are based on second- and higher-order terms in the expansion of the Markov chain generator. Our approximations achieve a higher degree of accuracy compared with diffusion approximations widely used for the last 50 years while retaining a similar computational complexity. To support our approximations, we present a combination of theoretical and numerical results across three different models. Our approximations are derived recursively through Stein/Poisson equations, and the theoretical results are proved using Stein’s method. Funding: X. Fang is partially supported by Hong Kong RGC [Grants 24301617, 14302418, and 14304917], a CUHK direct grant, and a CUHK start-up grant. J. G. Dai is partially supported by NSF [Grant CMMI-1537795]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2022.2362 .