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A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection

  • *Corresponding author: Huyên Pham

    *Corresponding author: Huyên Pham 

This work was supported by FiME (Finance for Energy Market Research Centre) and the "Finance et Développement Durable - Approches Quantitatives" EDF - CACIB Chair

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  • We consider the control of McKean-Vlasov dynamics (or mean-field control) with probabilistic state constraints. We rely on a level-set approach which provides a representation of the constrained problem in terms of an unconstrained one with exact penalization and running maximum or integral cost. The method is then extended to the common noise setting. Our work extends (Bokanowski, Picarelli, and Zidani, SIAM J. Control Optim. 54.5 (2016), pp. 2568–2593) and (Bokanowski, Picarelli, and Zidani, Appl. Math. Optim. 71 (2015), pp. 125–163) to a mean-field setting.

    The reformulation as an unconstrained problem is particularly suitable for the numerical resolution of the problem, that is achieved from an extension of a machine learning algorithm from (Carmona, Laurière, arXiv:1908.01613 to appear in Ann. Appl. Prob., 2019). A first application concerns the storage of renewable electricity in the presence of mean-field price impact and another one focuses on a mean-variance portfolio selection problem with probabilistic constraints on the wealth. We also illustrate our approach for a direct numerical resolution of the primal Markowitz continuous-time problem without relying on duality.

    Mathematics Subject Classification: Primary: 49N80, 49M99; Secondary: 68T07, 93E20.

    Citation:

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  • Figure 1.  Sample path of the controlled process $ X_t^\alpha $, with the analytical optimal control (for the unconstrained case) and the computed control. On the left figure we don't have the true control but plot the unconstrained one for comparison. Here $ \Lambda = 100 $

    Figure 2.  Histogram of $ X_T^\alpha $ for 50000 samples. Here $ \Lambda = 100 $

    Figure 3.  Auxiliary value function $ {\mathcal{Y}}_{\Lambda}(z) $ for several values of $ \Lambda $ in the constrained case, auxiliary value function $ {\mathcal{Y}}(z) $ in the unconstrained case

    Figure 4.  Conditional expectation $ E[X_t^\alpha\ |\ X_t^\alpha\leq 0.9] $ estimated with 50000 samples. The black line corresponds to $ \delta = 0.8 $. Here $ \Lambda = 100 $

    Figure 5.  Sample trajectory of the controlled process $ X^\alpha_t $ and the control for problem (28) (left). Variance $ \mathrm{Var}(X_t) $ estimated with 50000 samples for problem (28) (right) with $ \Lambda = 10 $

    Figure 6.  Auxiliary value function $ {\mathcal{U}}_{\Lambda}(z) $ for several values of $ \Lambda $

    Figure 7.  $ \hat w $ function value for the storage problem

    Figure 8.  Storage trajectories with the Level Set and Dynamic Programming method

    Table 1.  Estimate of the solution with maturity $ T = 1. $ Average and standard deviation observed over 10 independent runs are reported, with the relative error (in $ \% $). We also report the terminal expectation and variance of the approximated optimally controlled process for a single run. '$ \rm Not avail. $ ' means that we don't have a reference value and 'Tr.' means true. For problem (28), we take $ \Lambda = 10 $ and for problem (27) we illustrate the values obtained for $ \Lambda \in \{1., 10., 100\} $

    Problem Average Std Tr. val. Error
    (27) $ \Lambda = 1. $ -1.044 0.0010 $ \rm Not avail. $ $ \rm Not avail. $
    (27) $ \Lambda = 10. $ -1.044 0.0005 $ \rm Not avail. $ $ \rm Not avail. $
    27 $ \Lambda = 100. $ -1.045 0.0005 $ \rm Not avail. $ $ \rm Not avail. $
    (28) $ \Lambda = 10. $ -1.048 0.0017 -1.050 0.22
    (25) -1.050 0.0009 -1.050 0.07
    (25) [17] -1.052 0.0022 -1.050 0.13
    Problem $\mathbb{E}[X_T^{\alpha^*}]$ Tr. $\mathbb{E}[X_T^{\alpha^*}]$ $\mathrm{Var}(X_T^{\alpha^*})$ Tr. $\mathrm{Var}(X_T^{\alpha^*})$
    (27) $\Lambda = 1.$ 1.07 Not avail. 0.026 Not avail.
    (27) $\Lambda = 10.$ 1.07 Not avail. 0.026 Not avail.
    27 $\Lambda = 100.$ 1.07 Not avail. 0.027 Not avail.
    (28) $\Lambda = 10.$ 1.10 1.10 0.049 0.050
    (25) 1.10 1.10 0.050 0.050
    (25)[17] 1.10 1.10 0.053 0.050
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