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Optimal control of martingales in a radially symmetric environment

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  • Cox, Alexander M.G.
  • Robinson, Benjamin A.

Abstract

We study a stochastic control problem for continuous multidimensional martingales with fixed quadratic variation. In a radially symmetric environment, we are able to find an explicit solution to the control problem and find an optimal strategy. We show that it is optimal to switch between two strategies, depending only on the radius of the controlled process. The optimal strategies correspond to purely radial and purely tangential motion. It is notable that the value function exhibits smooth fit even when switching to tangential motion, where the radius of the optimal process is deterministic. Under sufficient regularity on the cost function, we prove optimality via viscosity solutions of a Hamilton–Jacobi–Bellman equation. We extend the results to cost functions that may become infinite at the origin. Extra care is required to solve the control problem in this case, since it is not clear how to define the optimal strategy with deterministic radius at the origin. Our results generalise some problems recently considered in Stochastic Portfolio Theory and Martingale Optimal Transport.

Suggested Citation

  • Cox, Alexander M.G. & Robinson, Benjamin A., 2023. "Optimal control of martingales in a radially symmetric environment," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 149-198.
    • Handle: RePEc:eee:spapps:v:159:y:2023:i:c:p:149-198
    DOI: 10.1016/j.spa.2023.01.016
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    1. Fausto Gozzi & Tiziano Vargiolu, 2002. "Superreplication of European multiasset derivatives with bounded stochastic volatility," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(1), pages 69-91, March.
    2. Larsson, Martin & Ruf, Johannes, 2021. "Relative arbitrage: sharp time horizons and motion by curvature," LSE Research Online Documents on Economics 108546, London School of Economics and Political Science, LSE Library.
    3. Fernholz, E. Robert & Karatzas, Ioannis & Ruf, Johannes, 2018. "Volatility and arbitrage," LSE Research Online Documents on Economics 75234, London School of Economics and Political Science, LSE Library.
    4. Lim, Tongseok, 2020. "Optimal martingale transport between radially symmetric marginals in general dimensions," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1897-1912.
    5. Martin Larsson & Johannes Ruf, 2021. "Relative arbitrage: Sharp time horizons and motion by curvature," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 885-906, July.
    6. Martin Larsson & Johannes Ruf, 2020. "Relative Arbitrage: Sharp Time Horizons and Motion by Curvature," Papers 2003.13601, arXiv.org, revised Feb 2021.
    7. A. Kyprianou & B. Surya, 2007. "Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels," Finance and Stochastics, Springer, vol. 11(1), pages 131-152, January.
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