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First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

ABSTRACT: In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

SUBMITTER: Bao J

PROVIDER: S-EPMC7897642 | biostudies-literature | 2021 Jan

REPOSITORIES: biostudies-literature

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First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

Bao Jianhai J Reisinger Christoph C Ren Panpan P Stockinger Wolfgang W

Proceedings. Mathematical, physical, and engineering sciences 20210106 2245

In this paper, we derive fully implementable first-order time-stepping schemes for McKean-Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean-Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong converge ...[more]

PMID: 33642922

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Project description:IntroductionConsensus guidelines recommend targeting a vancomycin area under the curve to minimum inhibitory concentration (AUC24 :MIC) ratio of 400-600 to improve therapeutic success and reduce nephrotoxicity. Although guidelines specify either Bayesian software or first-order equations may be used to estimate AUC24 , there are currently no large studies directly comparing these methods.ObjectiveTo compare calculated vancomycin AUC24 using first-order equations with two-drug concentrations at steady state to Bayesian two- and one-concentration estimations.MethodsThis was a single-center, retrospective cohort study of 978 adult hospitalized patients receiving intravenous vancomycin between 2017 and 2019. Patients were included if they received at least 72 h of vancomycin and had two-serum drug concentrations obtained. AUC24 was calculated using first-order analytic (linear), Bayesian two-concentration, and Bayesian one-concentration methods for each patient. The InsightRx™ software platform was used to calculate Bayesian AUC24 . Pearson's correlation and clinical agreement (based on AUC24 classified as subtherapeutic, therapeutic, or supratherapeutic) were used to assess agreement between methods. Bland-Altman plots were used to assess mean difference (MD) and 95% limits of agreement (LOA).ResultsExcellent agreement was observed between linear and Bayesian two-concentration methods (r = 0.963, clinical agreement = 87.4%) and Bayesian two-concentration and one-concentration methods (r = 0.931, clinical agreement = 88.5%); however, a degree of variability was noted with 95% LOA -99 to 76 (MD = -11.5 mg*h/L) and -92 to 113 (MD = -10.4 mg*h/L), for the respective comparisons. The agreement between linear and Bayesian one-concentration approaches was less than prior comparisons (r = 0.823, clinical agreement = 76.8%) and demonstrated the greatest amount of variability with 95% LOA -197 to 153 (MD = -21.9 mg*h/L).ConclusionsLinear and Bayesian two-concentration methods demonstrated high-level agreement with acceptable variability and may be considered comparable to estimate vancomycin AUC24 . As linear and Bayesian one-concentration methods demonstrated significant variability and suboptimal agreement, concerns exist surrounding the interchangeability of these methods in clinical practice, particularly at higher extremes of AUC24 .

Dataset Information

First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

Publications

First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems.

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