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Data-driven nonparametric robust control under dependence uncertainty

  • *Corresponding author: Erhan Bayraktar

    *Corresponding author: Erhan Bayraktar 

E. Bayraktar is supported by [National Science Foundation under grant DMS-2106556 and the Susan M. Smith chair]

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  • We consider a multi-period stochastic control problem where the multivariate driving stochastic factor of the system has known marginal distributions but uncertain dependence structure. To solve the problem, we propose to implement the nonparametric adaptive robust control framework. We aim to find the optimal control against the worst-case copulae in a sequence of shrinking uncertainty sets which are generated from continuously observing the data. Then, we use a stochastic gradient descent ascent algorithm to numerically handle the corresponding high dimensional dynamic inf-sup optimization problem. We present the numerical results in the context of utility maximization and show that the controller benefits from knowing more information about the uncertain model.

    Mathematics Subject Classification: Primary: 49L20, 49J55, 93E20, 93E35, 60G15, 65K05, 90C39, 90C40, 91G10, 91G60, 62G05.

    Citation:

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  • Figure 1.  Evolution of the wealth along out-of-sample paths. Upper left: box-plot of the wealth generated by $ \varphi^{*, \varepsilon} $; Upper right: box-plot of the wealth generated by $ \varphi^e $; Bottom left: box-plot of the wealth generated by $ \varphi^{\text{tr}} $; Bottom right: comparison of mean wealth

    Table 1.  Expected utility, variance, 30%-quantile, 90%-quantile, maximum, and minimum of the out-of-sample terminal wealth. AR: adaptive robust; TR: no uncertainty

    AR AR (No Marginals) TR
    $ V $ 19.8726 19.8659 19.8857
    $ \text{var}(X_T) $ 441.7334 628.7182 118.3159
    $ q_{0.30}(X_T) $ 92.4007 90.4748 96.3813
    $ q_{0.90}(X_T) $ 136.7146 144.7677 121.2001
    $ \text{max}(X_T) $ 190.4936 220.5724 144.6299
    $ \text{min}(X_T) $ 51.2641 48.7990 77.4599
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  • [1] E. Bayraktar and T. Chen, Nonparametric adaptive robust control under model uncertainty, Preprint, 2022. arXiv: 2202.10391.
    [2] D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Academic Press, 1978.
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    [14] P. Mohajerin Esfahani and D. Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Mathematical Programming, 171 (2018), 115-166.  doi: 10.1007/s10107-017-1172-1.
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