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.
Keywords: Uncertainty quantification, relative entropy, non-reversible diffusion processes, semi-Markov queueing models, stochastic control
@article{M2AN_2021__55_1_131_0,
author = {Birrell, Jeremiah and Katsoulakis, Markos A. and Rey-Bellet, Luc},
title = {Quantification of model uncertainty on path-space \protect\emph{via} goal-oriented relative entropy},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {131--169},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020070},
mrnumber = {4216833},
zbl = {1472.62042},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020070/}
}
TY - JOUR AU - Birrell, Jeremiah AU - Katsoulakis, Markos A. AU - Rey-Bellet, Luc TI - Quantification of model uncertainty on path-space via goal-oriented relative entropy JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 131 EP - 169 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020070/ DO - 10.1051/m2an/2020070 LA - en ID - M2AN_2021__55_1_131_0 ER -
%0 Journal Article %A Birrell, Jeremiah %A Katsoulakis, Markos A. %A Rey-Bellet, Luc %T Quantification of model uncertainty on path-space via goal-oriented relative entropy %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 131-169 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020070/ %R 10.1051/m2an/2020070 %G en %F M2AN_2021__55_1_131_0
Birrell, Jeremiah; Katsoulakis, Markos A.; Rey-Bellet, Luc. Quantification of model uncertainty on path-space via goal-oriented relative entropy. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 131-169. doi: 10.1051/m2an/2020070
[1] , On the representation of an integrated Gauss–Markov process. Sci. Math. Jpn. 77 (2015) 357–361. | MR | Zbl
[2] , An efficient finite difference method for parameter sensitivities of continuous time Markov chains. SIAM J. Numer. Anal. 50 (2012) 2237–2258. | MR | Zbl | DOI
[3] and , Optimal Control: Linear Quadratic Methods. Dover Books on Engineering. Dover Publications (2007).
[4] and , Goal-oriented sensitivity analysis for lattice kinetic Monte Carlo simulations. J. Chem. Phys. 140 (2014) 124108. | DOI
[5] , and , Fitting phase-type distributions via the EM algorithm. Scand. J. Stat. 23 (1996) 419–441. | Zbl
[6] , and , Robust bounds on risk-sensitive functionals via Rényi divergence. SIAM/ASA J. Uncertainty Quantif. 3 (2015) 18–33. | MR | Zbl | DOI
[7] , , and , Robust bounds and optimization at the large deviations scale for queueing models via Rényi divergence. Preprint (2020). | arXiv | MR
[8] and , Hypercontractivité do semi-groups de diffusion. C.R. Acad. Sci. Paris Sér I Math. 299 (1984) 775–778. | MR | Zbl
[9] and , Optimal controller/observer gains of discounted-cost LQG systems. Automatica 101 (2019) 471–474. | MR | Zbl | DOI
[10] and , Uncertainty quantification for markov processes via variational principles and functional inequalities. SIAM/ASA J. Uncertainty Quantif. 8 (2020) 539–572. | MR | Zbl | DOI
[11] and , Matrix-exponential distributions in applied probability. In: Probability Theory and Stochastic Modelling, Springer, New York (2017). | MR | Zbl | DOI
[12] , and , Concentration Inequalities. Oxford University Press, Oxford (2013). | MR | Zbl | DOI
[13] , and , Convex optimization. Berichte über verteilte messysteme, no. pt. 1, Cambridge University Press (2004).
[14] and , Measuring distribution model risk. Math. Finance 26 (2013) 395–411. | MR | Zbl | DOI
[15] and , Systematic stress tests with entropic plausibility constraints. J. Banking Finance 37 (2013) 1552–1559. | DOI
[16] , , , , , and , Statistical analysis of a telephone call center. J. Am. Stat. Assoc. 100 (2005) 36–50. | MR | Zbl | DOI
[17] and , Distinguishing and integrating aleatoric and epistemic variation in uncertainty quantification. ESAIM: M2AN 47 (2013) 635–662. | MR | Zbl | Numdam | DOI
[18] , On the distribution of the integrated square of the Ornstein-Uhlenbeck process. SIAM J. Appl. Math. 51 (1991) 568–574. | MR | Zbl | DOI
[19] and , A weak convergence approach to the theory of large deviations. Wiley Series in Probability and Statistics, John Wiley & Sons, New York (2011). | MR | Zbl
[20] , , and , Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics. SIAM/ASA J. Uncertainty Quantif. 4 (2016) 80–111. | MR | Zbl | DOI
[21] , , and , Sensitivity analysis for rare events based on Rényi divergence. Ann. Appl. Probab. 30 (2020) 1507–1533. | MR | Zbl | DOI
[22] and , The Basel II Risk Parameters: Estimation, Validation, Stress Testing – With Applications to Loan Risk Management. Springer, Berlin-Heidelberg (2011). | DOI
[23] , Examples of fitting structured phase-type distributions. Appl. Stochastic Models Data Anal. 10 (1994) 247–255. | Zbl | DOI
[24] , Functional integration and partial differential equations. In: Vol. 109 of Annals of Mathematics Studies. Princeton University Press (2016).
[25] and , On the entropy for semi-Markov processes. J. Appl. Probab. 40 (2003) 1060–1068. | MR | Zbl | DOI
[26] , Monte Carlo methods in financial engineering. In: Stochastic Modelling and Applied Probability, Springer, New York (2013). | MR | Zbl
[27] and , Robust risk measurement and model risk. Quant. Finance 14 (2014) 29–58. | MR | Zbl | DOI
[28] , Likelihood ratio gradient estimation for stochastic systems. Commun. ACM 33 (1990) 75–84. | DOI
[29] , and , 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. | MR | Zbl | DOI
[30] , , and , How biased is your model? Concentration inequalities, information and model bias. IEEE Trans. Inf. Theory 66 (2020) 3079–3097. | MR | Zbl | DOI
[31] and , A simple framework to justify linear response theory. Nonlinearity 23 (2010) 909–922. | MR | Zbl | DOI
[32] and , Stochastic equations for complex systems: theoretical and computational topics. In: Mathematical Engineering, Springer International Publishing (2015). | MR | Zbl | DOI
[33] and , Applied semi-Markov Processes. Springer, New York (2006). | MR | Zbl
[34] and , Brownian motion and stochastic calculus. in: Graduate Texts in Mathematics, Springer, New York (2014).
[35] , and , Scalable information inequalities for uncertainty quantification. J. Comput. Phys. 336 (2017) 513–545. | MR | Zbl | DOI
[36] , and , Spectral methods for parametric sensitivity in stochastic dynamical systems. Biophys. J. 92 (2007) 379–393. | DOI
[37] and , Numerical methods for stochastic control problems in continuous time. In: Stochastic Modelling and Applied Probability, Springer, New York (2013). | MR | Zbl
[38] , , and , Convergence properties of the nelder-mead simplex method in low dimensions. SIAM J. Optim. 9 (1998) 112–147. | MR | Zbl | DOI
[39] , Robust sensitivity analysis for stochastic systems. Math. Oper. Res. 41 (2016) 1248–1275. | MR | Zbl | DOI
[40] and , On divergences and informations in statistics and information theory. IEEE Trans. Inf. Theory 52 (2006) 4394–4412. | MR | Zbl | DOI
[41] and , Semi-Markov processes and reliability. In: Statistics for Industry and Technology, Birkhäuser Boston (2012). | MR | Zbl
[42] , A robust control framework for option pricing. Math. Oper. Res. 22 (1997) 202–221. | MR | Zbl | DOI
[43] and , A simplex method for function minimization. Comput. J. 7 (1965) 308–313. | MR | Zbl | DOI
[44] , , , and , Optimal uncertainty quantification. SIAM Rev. 55 (2013) 271–345. | MR | Zbl | DOI
[45] , Applications of mathematics in economics. MAA notes, Mathematical Association of America (2013). | MR | Zbl | DOI
[46] and , A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics. J. Chem. Phys. 138 (2013) 054115. | DOI
[47] and S.T. J, Uncertainty within economic models. In: World Scientific Series In Economic Theory, World Scientific Publishing Company (2014). | Zbl
[48] and , Efficient stochastic sensitivity analysis of discrete event systems. J. Comput. Phys. 221 (2007) 724–738. | MR | Zbl | DOI
[49] , Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations. J. Phys. Chem. B 110 (2006) 15063–15074. | DOI
[50] and , Irreversible Langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28 (2015) 2081–2103. | MR | Zbl | DOI
[51] , and , A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems. J. Chem. Phys. 136 (2012) 034115. | DOI
[52] , Stochastic Calculus for Finance II: Continuous-time Models. In: Vol. 11 of Springer Finance Textbooks, Springer, New York (2004). | MR | Zbl
[53] , A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri Poincare (B) Probab. Statistics 36 (2000) 435–445. | MR | Zbl | Numdam | DOI
Cité par Sources :
