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Linear–quadratic stochastic Volterra controls I: Causal feedback strategies

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  • Hamaguchi, Yushi
  • Wang, Tianxiao

Abstract

In this paper, we formulate and investigate the notion of causal feedback strategies arising in linear–quadratic control problems for stochastic Volterra integral equations (SVIEs) with singular and non-convolution-type coefficients. We show that there exists a unique solution, which we call the causal feedback solution, to the closed-loop system of a controlled SVIE associated with a causal feedback strategy. Furthermore, introducing two novel equations named a Type-II extended backward stochastic Volterra integral equation and a Lyapunov–Volterra equation, we prove a duality principle and a representation formula for a quadratic functional of controlled SVIEs in the framework of causal feedback strategies.

Suggested Citation

  • Hamaguchi, Yushi & Wang, Tianxiao, 2024. "Linear–quadratic stochastic Volterra controls I: Causal feedback strategies," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:spapps:v:176:y:2024:i:c:s0304414924001558
    DOI: 10.1016/j.spa.2024.104449
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    References listed on IDEAS

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    1. Abi Jaber, Eduardo & El Euch, Omar, 2019. "Markovian structure of the Volterra Heston model," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 63-72.
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    4. Eduardo Abi Jaber & Enzo Miller & Huyen Pham, 2021. "Integral Operator Riccati Equations Arising in Stochastic Volterra Control Problems," Post-Print hal-03264893, HAL.
    5. Nacira Agram & Bernt Øksendal, 2015. "Malliavin Calculus and Optimal Control of Stochastic Volterra Equations," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1070-1094, December.
    6. Eduardo Abi Jaber & Omar El Euch, 2019. "Markovian structure of the Volterra Heston model," Post-Print hal-01716696, HAL.
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