Quantification of model uncertainty on path-space via goal-oriented relative entropy. (English) Zbl 1472.62042
Summary: Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g., for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.
MSC:
| 62F35 | Robustness and adaptive procedures (parametric inference) |
| 62B10 | Statistical aspects of information-theoretic topics |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60J60 | Diffusion processes |
| 60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |
| 93E20 | Optimal stochastic control |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 94A17 | Measures of information, entropy |
Keywords:
uncertainty quantification; relative entropy; non-reversible diffusion processes; semi-Markov queueing models; stochastic controlReferences:
| [1] | M. Abundo, On the representation of an integrated Gauss-Markov process. Sci. Math. Jpn. 77 (2015) 357-361. [Google Scholar] · Zbl 1336.60154 |
| [2] | D. Anderson, An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM J. Numer. Anal. 50 (2012) 2237-2258. [Google Scholar] · Zbl 1264.60050 |
| [3] | B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods. Dover Books on Engineering. Dover Publications (2007). [Google Scholar] |
| [4] | G. Arampatzis and M.A. Katsoulakis, Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations. J. Chem. Phys. 140 (2014) 124108. [Google Scholar] · Zbl 1320.65004 |
| [5] | S. Asmussen, O. Nerman and M. Olsson, Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23 (1996) 419-441. [Google Scholar] · Zbl 0898.62104 |
| [6] | R. Atar, K. Chowdhary and P. Dupuis, Robust bounds on risk-sensitive functionals via Rényi divergence. SIAM/ASA J. Uncertainty Quantif. 3 (2015) 18-33. [Google Scholar] · Zbl 1341.60008 |
| [7] | R. Atar, A. Budhiraja, P. Dupuis and R. Wu, Robust bounds and optimization at the large deviations scale for queueing models via Rényi divergence. Preprint (2020). [Google Scholar] |
| [8] | D. Bakry and M. Emery, Hypercontractivité do semi-groups de diffusion. C.R. Acad. Sci. Paris Sér I Math. 299 (1984) 775-778. [Google Scholar] · Zbl 0563.60068 |
| [9] | H. Bijl and T.B. Schön, Optimal controller/observer gains of discounted-cost LQG systems. Automatica 101 (2019) 471-474. [Google Scholar] · Zbl 1412.49067 |
| [10] | J. Birrell and L. Rey-Bellet, Uncertainty quantification for markov processes via variational principles and functional inequalities. SIAM/ASA J. Uncertainty Quantif. 8 (2020) 539-572. [Google Scholar] · Zbl 1501.47073 |
| [11] | M. Bladt and B.F. Nielsen, Matrix-exponential distributions in applied probability. In: Probability Theory and Stochastic Modelling, Springer, New York (2017). [Google Scholar] · Zbl 1375.60002 |
| [12] | S. Boucheron, G. Lugosi and P. Massart, Concentration Inequalities. Oxford University Press, Oxford (2013). [Google Scholar] · Zbl 1279.60005 |
| [13] | S. Boyd, S.P. Boyd and L. Vandenberghe, Convex optimization. Berichte über verteilte messysteme, no. pt. 1, Cambridge University Press (2004). [Google Scholar] · Zbl 1058.90049 |
| [14] | T. Breuer and I. Csiszár, Measuring distribution model risk. Math. Finance 26 (2013) 395-411. [Google Scholar] · Zbl 1348.91290 |
| [15] | T. Breuer and I. Csiszár, Systematic stress tests with entropic plausibility constraints. J. Banking Finance 37 (2013) 1552-1559. [Google Scholar] |
| [16] | L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn and L. Zhao, Statistical analysis of a telephone call center. J. Am. Stat. Assoc. 100 (2005) 36-50. [Google Scholar] · Zbl 1117.62303 |
| [17] | K. Chowdhary and P. Dupuis, Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification. ESAIM: M2AN 47 (2013) 635-662. [Google Scholar] · Zbl 1266.65009 |
| [18] | T. Dankel, On the distribution of the integrated square of the Ornstein-Uhlenbeck process. SIAM J. Appl. Math. 51 (1991) 568-574. [Google Scholar] · Zbl 0721.60043 |
| [19] | P. Dupuis and R.S. Ellis, A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics, John Wiley & Sons, New York (2011). [Google Scholar] · Zbl 0904.60001 |
| [20] | P. Dupuis, M.A. Katsoulakis, Y. Pantazis and P. Plecháč, Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics. SIAM/ASA J. Uncertainty Quantif. 4 (2016) 80-111. [Google Scholar] · Zbl 1371.65004 |
| [21] | P. Dupuis, M.A. Katsoulakis, Y. Pantazis and L. Rey-Bellet, Sensitivity analysis for rare events based on Rényi divergence. Ann. Appl. Probab. 30 (2020) 1507-1533. [Google Scholar] · Zbl 1464.60024 |
| [22] | B. Engelmann and R. Rauhmeier, The Basel II Risk Parameters: Estimation, Validation, Stress Testing - With Applications to Loan Risk Management. Springer, Berlin-Heidelberg (2011). [Google Scholar] |
| [23] | M.J. Faddy, Examples of fitting structured phase-type distributions. Appl. Stochastic Models Data Anal. 10 (1994) 247-255. [Google Scholar] · Zbl 0822.60085 |
| [24] | M.I. Freidlin, Functional integration and partial differential equations. In: Vol. 109 of Annals of Mathematics Studies. Princeton University Press (2016). [Google Scholar] · Zbl 0568.60057 |
| [25] | V. Girardin and N. Limnios, On the entropy for semi-Markov processes. J. Appl. Probab. 40 (2003) 1060-1068. [Google Scholar] · Zbl 1057.60081 |
| [26] | P. Glasserman, Monte Carlo methods in financial engineering. In: Stochastic Modelling and Applied Probability, Springer, New York (2013). [Google Scholar] · Zbl 1038.91045 |
| [27] | P. Glasserman and X. Xu, Robust risk measurement and model risk. Quant. Finance 14 (2014) 29-58. [Google Scholar] · Zbl 1294.91076 |
| [28] | P.W. Glynn, Likelihood ratio gradient estimation for stochastic systems. Commun. ACM 33 (1990) 75-84. [Google Scholar] |
| [29] | K. Gourgoulias, M.A. Katsoulakis and L. Rey-Bellet, Information metrics for long-time errors in splitting schemes for stochastic dynamics and parallel kinetic Monte Carlo. SIAM J. Sci. Comput. 38 (2016) A3808-A3832. [Google Scholar] · Zbl 1355.65006 |
| [30] | K. Gourgoulias, M.A. Katsoulakis, L. Rey-Bellet and J. Wang, How biased is your model? Concentration inequalities, information and model bias. IEEE Trans. Inf. Theory 66 (2020) 3079-3097. [Google Scholar] · Zbl 1448.94102 |
| [31] | M. Hairer and A.J. Majda, A simple framework to justify linear response theory. Nonlinearity 23 (2010) 909-922. [Google Scholar] · Zbl 1186.82006 |
| [32] | S. Heinz and H. Bessaih, Stochastic equations for complex systems: theoretical and computational topics. In: Mathematical Engineering, Springer International Publishing (2015). [Google Scholar] · Zbl 1319.60003 |
| [33] | J. Janssen and R. Manca, Applied semi-Markov Processes. Springer, New York (2006). [Google Scholar] · Zbl 1096.60002 |
| [34] | I. Karatzas and S. Shreve, Brownian motion and stochastic calculus. in: Graduate Texts in Mathematics, Springer, New York (2014). [Google Scholar] · Zbl 0638.60065 |
| [35] | M.A. Katsoulakis, L. Rey-Bellet and J. Wang, Scalable information inequalities for uncertainty quantification. J. Comput. Phys. 336 (2017) 513-545. [Google Scholar] · Zbl 1375.82043 |
| [36] | D. Kim, B.J. Debusschere and H.N. Najm, Spectral methods for parametric sensitivity in stochastic dynamical systems. Biophys. J. 92 (2007) 379-393. [PubMed] [Google Scholar] |
| [37] | H. Kushner and P.G. Dupuis, Numerical methods for stochastic control problems in continuous time. In: Stochastic Modelling and Applied Probability, Springer, New York (2013). [Google Scholar] · Zbl 0968.93005 |
| [38] | J.C. Lagarias, J.A. Reeds, M.H. Wright and P.E. Wright, Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J. Optim. 9 (1998) 112-147. [Google Scholar] · Zbl 1005.90056 |
| [39] | H. Lam, Robust sensitivity analysis for stochastic systems. Math. Oper. Res. 41 (2016) 1248-1275. [Google Scholar] · Zbl 1361.65008 |
| [40] | F. Liese and I. Vajda, On divergences and informations in statistics and information theory. IEEE Trans. Inf. Theory 52 (2006) 4394-4412. [Google Scholar] · Zbl 1287.94025 |
| [41] | N. Limnios and G. Oprisan, Semi-Markov processes and reliability. In: Statistics for Industry and Technology, Birkhäuser Boston (2012). [Google Scholar] · Zbl 0990.60004 |
| [42] | W.M. McEneaney, A robust control framework for option pricing. Math. Oper. Res. 22 (1997) 202-221. [Google Scholar] · Zbl 0871.90010 |
| [43] | J.A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7 (1965) 308-313. [Google Scholar] · Zbl 0229.65053 |
| [44] | H. Owhadi, C. Scovel, T.J. Sullivan, M. McKerns and M. Ortiz, Optimal uncertainty quantification. SIAM Rev. 55 (2013) 271-345. [Google Scholar] · Zbl 1278.60040 |
| [45] | W. Page, Applications of mathematics in economics. MAA notes, Mathematical Association of America (2013). [Google Scholar] · Zbl 1270.91006 |
| [46] | Y. Pantazis and M.A. Katsoulakis, A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics. J. Chem. Phys. 138 (2013) 054115. [Google Scholar] |
| [47] | H.L. Peter and S.T. J, Uncertainty within economic models. In: World Scientific Series In Economic Theory, World Scientific Publishing Company (2014). [Google Scholar] |
| [48] | S. Plyasunov and A.P. Arkin, Efficient stochastic sensitivity analysis of discrete event systems. J. Comput. Phys. 221 (2007) 724-738. [Google Scholar] · Zbl 1107.65301 |
| [49] | H. Qian, Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations. J. Phys. Chem. B 110 (2006) 15063-15074. [PubMed] [Google Scholar] · doi:10.1021/jp061858z |
| [50] | L. Rey-Bellet and K. Spiliopoulos, Irreversible Langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28 (2015) 2081-2103. [Google Scholar] · Zbl 1338.60086 |
| [51] | P.W. Sheppard, M. Rathinam and M. Khammash, A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems. J. Chem. Phys. 136 (2012) 034115. [Google Scholar] |
| [52] | S.E. Shreve, Stochastic Calculus for Finance II: Continuous-time Models. In: Vol. 11 of Springer Finance Textbooks, Springer, New York (2004). [Google Scholar] · Zbl 1068.91041 |
| [53] | L. Wu, A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri Poincare (B) Probab. Statistics 36 (2000) 435-445. [Google Scholar] · Zbl 0972.60003 |
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