Approximation of the invariant distribution for a class of ergodic jump diffusions

Gloter, Arnaud; Honoré, Igor; Loukianova, Dasha

doi:10.1051/ps/2020023

Cited by 2 publications

(2 citation statements)

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“…There have been many results concerning the error estimates between the ergodic measures of (Y t ) t 0 and ( Ỹn ) n∈N 0 , see for instance [5,9,11,20,21,37,38,39,44,49], but most of them are asymptotic type. For asymptotic results, we recall those in the literatures [11,28,44] whose settings are close to ours. [28,44] considered an empirical measure Π n em of π em for a class of SDEs driven by multiplicative Lévy noises, showing that…”

Section: Introductionmentioning

confidence: 71%

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An approximation to steady-state of M/Ph/n+M queue

Jin¹,

Pang²,

Xu³

et al. 2021

Preprint

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In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the M/P h/n + M queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the prelimit diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property.To establish the error bound, we employ the recently developed Stein's equation and Malliavian calculus for multi-dimensional diffusions. The main difficulty lies in the nondifferentiability of the drift in the limiting diffusion, so that the standard approaches in Euler type of schemes for diffusions and Stein's method do not work. We first propose a mollified diffusion which has a sufficiently smooth drift to circumvent the nondifferentiability difficulty. We then provide some insights on the limiting diffusion and approximate diffusion from the algorithm by investigating some useful occupation times, as well as the associated Harnack inequalities, the support of the invariant measures and ergodicity properties. These results are used in analyzing the Stein's equation, which provide useful estimates to bound the differences between the corresponding invariant measures. CONTENTS1. Introduction 1.1. Summary of results and contributions 1.2. Related works 1.3. Organization of the paper 1.4. Notations 2. Proof of the main result 3. Malliavin calculus for SDE (1.6) and Stein's equation 3.1. Malliavin calculus associated with SDE (1.6) 3.2. Stein's equation of SDE (1.6) 4. Estimates for weighted occupation times: important ingredients 5. Proofs of Propositions 2.3 and 2.4 Appendix A. Harnack inequality Appendix B. Proof of Proposition 2.1 Appendix C. Proof of Proposition 2.2 Appendix D. Estimates for Jacobi flow and Malliavin derivative

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Section: Introductionmentioning

confidence: 71%

“…For asymptotic results, we recall those in the literatures [11,28,44] whose settings are close to ours. [28,44] considered an empirical measure Π n em of π em for a class of SDEs driven by multiplicative Lévy noises, showing that…”

Section: Introductionmentioning

confidence: 71%

An approximation to steady-state of M/Ph/n+M queue

Jin¹,

Pang²,

Xu³

et al. 2021

Preprint

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An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

Jin,

Pang,

et al. 2024

Mathematics of OR

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In this paper, we develop a stochastic algorithm based on the Euler–Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of [Formula: see text] queues in the Halfin–Whitt regime. Specifically, we prove a nonasymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein’s method for multidimensional diffusions, in which the regularity of Stein’s equation obtained by the partial differential equation (PDE) theory plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of [Formula: see text] queues and its Euler–Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein’s equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role. Funding: X. Jin is supported in part by the Fundamental Research Funds for the Central Universities [Grants JZ2022HGQA0148 and JZ2023HGTA0170]. G. Pang is supported in part by the U.S. National Science Foundation [Grants DMS-1715875 and DMS-2216765]. L. Xu is supported in part by the National Nature Science Foundation of China [Grant 12071499], Macao Special Administrative Region [Grant FDCT 0090/2019/A2], and the University of Macau [Grant MYRG2018-00133-FST]. This work was supported by U.S. National Science Foundation [Grant DMS-2108683].

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Approximation of the invariant distribution for a class of ergodic jump diffusions

Cited by 2 publications

References 16 publications

An approximation to steady-state of M/Ph/n+M queue

An approximation to steady-state of M/Ph/n+M queue

An Approximation to the Invariant Measure of the Limiting Diffusion of G/Ph/n + GI Queues in the Halfin–Whitt Regime and Related Asymptotics

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