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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.


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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
     | Show Table
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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.
    [3] T. Bhudisaksang and Á. Cartea, Online drift estimation for jump-diffusion processes, Bernoulli - Journal of the Bernoulli Society, 27 (2021), 2494-2518.  doi: 10.3150/20-BEJ1319.
    [4] T. BieleckiT. Chen and I. Cialenco, Recursive construction of confidence regions, Electron. J. Statist., 11 (2017), 4674-4700.  doi: 10.1214/17-EJS1362.
    [5] T. R. Bielecki, T. Chen and I. Cialenco, Time-inconsistent Markovian control problems under model uncertainty with application to the mean-variance portfolio selection, International Journal of Theoretical and Applied Finance, 24 (2021), Paper No. 2150003, 28 pp. doi: 10.1142/S0219024921500035.
    [6] H. F. Chen and L. Guo, Identification and Stochastic Adaptive Control, Systems & Control: Foundations & Applications. Birkhäuser Boston, Inc., 1991. doi: 10.1007/978-1-4612-0429-9.
    [7] T. Chen and M. Ludkovski, A machine learning approach to adaptive robust utility maximization and hedging, SIAM Journal on Financial Mathematics, 12 (2021), 1226-1256.  doi: 10.1137/20M1336023.
    [8] T. Chen and J. Myung, Nonparametric adaptive bayesian stochastic control under model uncertainty, Preprint, 2020. arXiv: 2011.04804.
    [9] E. Del BarrioE. Giné and C. Matrán, Central limit theorems for the wasserstein distance between the empirical andthe true distributions, The Annals of Probability, 27 (1999), 1009-1071.  doi: 10.1214/aop/1022677394.
    [10] N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.  doi: 10.1007/s00440-014-0583-7.
    [11] R. Gao and A. Kleywegt, Data-driven robust optimization with known marginal distributions, In Technical Report, Georgia Institute of Technology, 2017.
    [12] I. Gilboa and D. Schmeidler., Maxmin expected utility with nonunique prior, J. Math. Econom., 18 (1989), 141-153.  doi: 10.1016/0304-4068(89)90018-9.
    [13] P. R. Kumar and P. Varaiya, Stochastic Systems: Estimation, Identification, and Adaptive Control, Classics in Applied Mathematics, 75. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016. doi: 10.1137/1.9781611974263.
    [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.
    [15] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning, The MIT Press, 2006.
    [16] U. Rieder, Bayesian dynamic programming, Adv. Appl. Prob., 7 (1975), 330-348.  doi: 10.2307/1426080.
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