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Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations. (English) Zbl 1436.62077

Author’s abstract: Multi-dimensional stochastic differential equations (SDEs) are a powerful tool to describe dynamics of phenomena that change over time. We focus on the parametric estimation of such SDEs based on partial observations when only a one-dimensional component of the system is observable. We consider two families of SDE, the elliptic family with a full-rank diffusion coefficient and the hypoelliptic family with a degenerate diffusion coefficient. Estimation for the second class is much more difficult and only few estimation methods have been proposed. Here, we adopt the framework of the optimal control theory to derive a contrast (or cost function) based on the best control sequence mimicking the (unobserved) Brownian motion. We propose a full data-driven approach to estimate the drift and diffusion coefficient parameters. Numerical simulations made on different examples (Harmonic Oscillator, FitzHugh-Nagumo, Lotka-Volterra) reveal our method produces good pointwise estimate for an acceptable computational price with, interestingly, no performance drop for hypoelliptic systems.

MSC:

62F10 Point estimation
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
49K20 Optimality conditions for problems involving partial differential equations
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[1] Aït-Sahalia, Y., Closed-form likelihood expansions for multivariate diffusions, Ann Stat, 36, 906-937 (2008) · Zbl 1246.62180
[2] Bally, V.; Talay, D., The law of the Euler scheme for stochastic differential equations, Probability Theory and Related Fields, 104, 1, 43-60 (1996) · Zbl 0838.60051
[3] Bertsekas, D., Dynamic programming and optimal control (2005), Belmont: Athena Scientific, Belmont · Zbl 1125.90056
[4] Beskos, A.; Papaspiliopoulos, O.; Roberts, Go; Fearnhead, P., Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes, J R Stat Soc Ser B Stat Methodol, 68, 333-382 (2006) · Zbl 1100.62079
[5] Bibby, B.; Sorensen, M., Martingale estimating functions for discretely observed diffusion processes, Bernoulli, 1, 17-39 (1995) · Zbl 0830.62075
[6] Bierkens J, Van der Meulen F, Schauer M (2018) Simulation of elliptic and hypo-elliptic conditional diffusions. arXiv preprint arXiv:1810.01761 · Zbl 1448.60164
[7] Brunel, Nj-B; Clairon, Q., A tracking approach to parameter estimation in linear ordinary differential equations, Electron J Stat, 9, 2903-2949 (2015) · Zbl 1330.62168
[8] Cattiaux, Patrick; León, José R.; Prieur, Clémentine, Estimation for stochastic damping hamiltonian systems under partial observation—I. Invariant density, Stochastic Processes and their Applications, 124, 3, 1236-1260 (2014) · Zbl 1279.62082
[9] Cattiaux P, León J, Prieur C (2014b) Estimation for stochastic damping hamiltonian systems under partial observation. II. Drift term. ALEA 11:359-384 · Zbl 1346.62066
[10] Çimen, Tayfun; Banks, Stephen P., Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria, Systems & Control Letters, 53, 5, 327-346 (2004) · Zbl 1157.49313
[11] Çimen, Tayfun; Banks, Stephen P., Nonlinear optimal tracking control with application to super-tankers for autopilot design, Automatica, 40, 11, 1845-1863 (2004) · Zbl 1059.93505
[12] Clairon, Q.; Brunel, N., Optimal control and additive perturbations help in estimating ill-posed and uncertain dynamical systems, J Am stat Assoc, 113, 523, 1195-1209 (2018) · Zbl 1402.62031
[13] Clairon, Q.; Brunel, N., Tracking for parameter and state estimation in possibly misspecified partially observed linear ordinary differential equations, J stat Plan Inference, 199, 188-206 (2018) · Zbl 1421.62020
[14] Comte, F.; Prieur, C.; Samson, A., Adaptive estimation for stochastic damping Hamiltonian systems under partial observation, Stoch Process Appl, 127, 11, 3689-3718 (2017) · Zbl 1494.62015
[15] Cuenod, C.; Favetto, B.; Genon-Catalot, V.; Rozenholc, Y.; Samson, A., Parameter estimation and change-point detection from dynamic contrast enhanced MRI data using stochastic differential equations, Math Biosci, 233, 68-76 (2011) · Zbl 1226.92046
[16] Dietz, H., Asymptotic behaviour of trajectory fitting estimators for certain non-ergodic sde, Stat Inference Stoch Process, 4, 249-258 (2001) · Zbl 1006.62068
[17] Ditlevsen, S.; Greenwood, P., The morris-lecar neuron model embeds a leaky integrate-and-fire model, J Math Biol, 67, 239-259 (2013) · Zbl 1315.60044
[18] Ditlevsen, S.; Samson, A., Estimation in the partially observed stochastic Morris-Lecar neuronal model with particle filter and stochastic approximation methods, Ann Appl Stat, 2, 674-702 (2014) · Zbl 1454.62246
[19] Ditlevsen S, Samson A (2017) Hypoelliptic diffusions: discretization, filtering and inference from complete and partial information, pp 1-33. arXiv:1707.04235v1
[20] Ditlevsen, S.; Yip, K.; Holstein-Rathlou, N., Parameter estimation in a stochastic model of the tubuloglomerular feedback mechanism in a rat nephron, Math Biosci, 194, 49-69 (2005) · Zbl 1063.92019
[21] Durham, Gb; Gallant, Ar, Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes, J Bus Econ Stat, 20, 297-338 (2002)
[22] Elerian, O.; Chib, S.; Shephard, N., Likelihood inference for discretely observed nonlinear diffusions, Econometrica, 69, 959-993 (2001) · Zbl 1017.62068
[23] Eraker, B., MCMC analysis of diffusion models with application to finance, J Bus Econ Stat, 19, 177-191 (2001)
[24] Fitzhugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophys J, 6, 445-466 (1961)
[25] Gerstner, W.; Kistler, W., Spiking neuron models (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1100.92501
[26] Gloter, A., Parameter estimation for a discretely observed integrated diffusion process, Scand J Stat, 33, 83-104 (2006) · Zbl 1126.62070
[27] Golightly, A.; Wilkinson, Dj, Bayesian sequential inference for nonlinear multivariate diffusions, Stat Comput, 16, 323-338 (2006)
[28] Golightly, A.; Wilkinson, Dj, Bayesian inference for nonlinear multivariate diffusion models observed with error, Comput Stat Data Anal, 52, 1674-1693 (2008) · Zbl 1452.62603
[29] Graham M, Storkey A (2017) Asymptotically exact inference in differenciable generative models, p 14. arXiv:1605.07826 · Zbl 1380.65025
[30] Iolov, A.; Ditlevsen, S.; Longtin, A., Optimal design for estimation in diffusion processes from first hitting times, SIAM J Uncertain Quantif, 5, 88-110 (2017) · Zbl 1365.62305
[31] Ionides, El; Bhadra, A.; Atchade, Y.; King, Aa, Iterated filtering, Ann Stat, 39, 1776-1802 (2011) · Zbl 1220.62103
[32] Ionides, El; Breto, C.; King, Aa, Inference for nonlinear dynamical systems, Proc Natl Acad Sci, 103, 18438-18443 (2006)
[33] Jensen, Ac; Ditlevsen, S.; Kessler, M.; Papaspiliopoulos, O., Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model, Phys Rev E, 86, 041114 (2012)
[34] Kessler, M., Estimation of an ergodic diffusion from discrete observations, Scand J Stat, 24, 211-229 (1997) · Zbl 0879.60058
[35] King, Aa; Nguyen, D.; Ionides, El, Statistical inference for partially observed markov processes via the R package pomp, J Stat Softw, 69, 1-43 (2016)
[36] Kloeden, P.; Platen, E., Numerical solution of stochastic differential equations (1992), Berlin: Springer, Berlin · Zbl 0925.65261
[37] Kutoyants, Y., Minimum distance parameter estimation for diffusion type observation, C R Acad Sci, 312, 637 (1991) · Zbl 0717.62075
[38] Leon, J.; Rodriguez, L.; Ruggiero, R., Consistency of a likelihood estimator for stochastic damping hamiltonian systems, Totally observed data. ESAIM PS, 23, 1-36 (2019) · Zbl 1415.62063
[39] Lipster, R.; Shiryaev, A., Statistics of random processes I: general theory (2001), Berlin: Springer, Berlin · Zbl 1008.62072
[40] Lotka, A., Elements of mathematical biology (1925), New York: Dover, New York · JFM 51.0416.06
[41] Mao, X.; Marion, G.; Renshaw, E., Environmental brownian noise suppresses explosions in populations dynamics, Stoch Process Appl, 97, 95-110 (2002) · Zbl 1058.60046
[42] Martin, C.; Sun, S.; Egerstedt, M., Optimal control, statistics and path planning, Math Comput Model, 33, 237-253 (2001) · Zbl 0976.65057
[43] Mattingly, J.; Stuart, Am; Higham, D., Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stoch Process Appl, 101, 185-232 (2002) · Zbl 1075.60072
[44] Meeds E, Welling M (2015) Optimization Monte Carlo: efficient and embarrassingly parallel likelihood-free inference. In: Cortes CN, Lawrence D, Lee DD, Sugiyama M, Garnett R (eds) Advances in neural information processing systems. Curran Associates, Inc, pp. 2080-2088. http://papers.nips.cc/paper/5881-optimization-monte-carlo-efficient-and-embarrassingly-parallellikelihood-free-inference.pdf
[45] Melnykova A (2019) Parametric inference for multidimensional hypoelliptic ergodic diffusion with full observations. hal-01704010v2 · Zbl 1465.62047
[46] Nagumo, J.; Animoto, S.; Yoshizawa, S., An active pulse transmission line simulating nerve axon, Proc Inst Radio Eng, 50, 2061-2070 (1962)
[47] Paninski, L.; Ahmadian, Y.; Ferreira, Dg; Koyama, S.; Rad, Kr; Vidne, M.; Vogelstein, J.; Wu, W., A new look at state-space models for neural data, J Comput Neurosci, 29, 107-126 (2010) · Zbl 1446.92161
[48] Paninski, L.; Vidne, M.; Depasquale, B.; Fereira, D., Inferring synaptic inputs given a noisy voltage trace via sequential Monte Carlo methods, J Comput Neurosci, 33, 1-19 (2012) · Zbl 1446.92162
[49] Pedersen, A., A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scand J Stat, 22, 55-71 (1995) · Zbl 0827.62087
[50] Pokern, Y.; Stuart, A.; Wiberg, P., Parameter estimation for partially observed hypoelliptic diffusions, J R Stat Soc B, 71, 49-73 (2009) · Zbl 1231.62152
[51] Pontryagin, Ls; Boltyanskii, Vg; Gamkrelidze, Rv; Mischenko, Ef, The mathematical theory of optimal processes (1962), Hoboken: Wiley-Interscience, Hoboken · Zbl 0102.32001
[52] Samson, A.; Thieullen, M., A contrast estimator for completely or partially observed hypoelliptic diffusion, Stoch Process Appl, 122, 2521-2552 (2012) · Zbl 1242.62092
[53] Sontag, E., Mathematical control theory: deterministic finite-dimensional systems (1998), New York: Springer, New York · Zbl 0945.93001
[54] Sørensen, H., Parametric inference for diffusion processes observed at discrete points in time: a survey, Int Stat Rev, 72, 337-354 (2004)
[55] Trelat E (2005) Controle optimal: theories et applications. Vuibert · Zbl 1112.49001
[56] Van der Meulen F, Schauer M (2016) Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Electron J Stat 11(2017):2358-2396 · Zbl 1378.62050
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