Adaptative design for statistical estimation with ergodic controls for Ornstein-Uhlenbeck processes

Alexandre Brouste; Youssef Esstafa; Alexandre Popier

doi:10.3934/mcrf.2025061

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2025. Doi: 10.3934/mcrf.2025061

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Adaptative design for statistical estimation with ergodic controls for Ornstein-Uhlenbeck processes

1.
Laboratoire Manceau de Mathématiques, Le Mans Université, France

^*Corresponding author: Alexandre Brouste
^*Corresponding author: Alexandre Brouste

Received: October 2025

Revised: October 2025

Early access: October 22, 2025

Published online: October 22, 2025

The authors are supported by the ANR project 'Efficient inference for large and high-frequency data' (ANR-21-CE40-0021) and the 'Efficience et Sobriété Numériques' research program under the aegis of the Institut Louis Bachelier, a joint initiative by Le Mans Université and EREN Groupe. The second co-author acknowledges support from the Etoiles Montantes en Pays de la Loire project, funded by the Pays de la Loire Region.

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Abstract

In this paper, we investigate the optimal design problem for asymptotically efficient estimation in the context of controlled Ornstein–Uhlenbeck processes. The study builds a connection between statistical optimal design and ergodic control theories. We focus on the estimation of an unknown drift parameter with continuous observations and the possibility of an additive admissible control. Two notions of statistical asymptotic efficiency are examined in the controlled setting. The first relies on maximizing the Fisher information with respect to controls on finite horizons, while the second is based on ergodic (infinite-horizon) controls. We establish that both definitions lead to the same asymptotic variance in the case of deterministic controls, and also provide results for stochastic controls.

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Mathematics Subject Classification: Primary: 62K05, 62M05; Secondary: 49N99, 60G99.

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