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Exploratory Randomization for Discrete-Time Linear Exponential Quadratic Gaussian (LEQG) Problem

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  • Sebastien Lleo
  • Wolfgang Runggaldier

Abstract

We investigate exploratory randomization for an extended linear-exponential-quadratic-Gaussian (LEQG) control problem in discrete time. This extended control problem is related to the structure of risk-sensitive investment management applications. We introduce exploration through a randomization of the control. Next, we apply the duality between free energy and relative entropy to reduce the LEQG problem to an equivalent risk-neutral LQG control problem with an entropy regularization term, see, e.g. Dai Pra et al. (1996), for which we present a solution approach based on Dynamic Programming. Our approach, based on the energy-entropy duality may also be considered as leading to a justification for the use, in the literature, of an entropy regularization when applying a randomized control.

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  • Sebastien Lleo & Wolfgang Runggaldier, 2025. "Exploratory Randomization for Discrete-Time Linear Exponential Quadratic Gaussian (LEQG) Problem," Papers 2501.06275, arXiv.org.
    • Handle: RePEc:arx:papers:2501.06275
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    References listed on IDEAS

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    1. Mark Davis & SEBastien Lleo, 2008. "Risk-sensitive benchmarked asset management," Quantitative Finance, Taylor & Francis Journals, vol. 8(4), pages 415-426.
    2. Davis, Mark & Lleo, Sébastien, 2020. "Debiased expert forecasts in continuous-time asset allocation," Journal of Banking & Finance, Elsevier, vol. 113(C).
    3. Lleo, Sébastien & Runggaldier, Wolfgang J., 2024. "On the separation of estimation and control in risk-sensitive investment problems under incomplete observation," European Journal of Operational Research, Elsevier, vol. 316(1), pages 200-214.
    4. Ben Hambly & Renyuan Xu & Huining Yang, 2020. "Policy Gradient Methods for the Noisy Linear Quadratic Regulator over a Finite Horizon," Papers 2011.10300, arXiv.org, revised Jun 2021.
    5. Nicole Bäuerle & Ulrich Rieder, 2014. "More Risk-Sensitive Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 105-120, February.
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    7. Mark H.A. Davis & Sébastien Lleo, 2021. "Risk‐sensitive benchmarked asset management with expert forecasts," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1162-1189, October.
    8. Yanwei Jia & Xun Yu Zhou, 2022. "q-Learning in Continuous Time," Papers 2207.00713, arXiv.org, revised Apr 2023.
    9. Yanwei Jia & Xun Yu Zhou, 2021. "Policy Evaluation and Temporal-Difference Learning in Continuous Time and Space: A Martingale Approach," Papers 2108.06655, arXiv.org, revised Feb 2022.
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