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January  2020, 5: 4 doi: 10.1186/s41546-020-00046-x

Uncertainty and filtering of hidden Markov models in discrete time

Samuel N. Cohen

Mathematical Institute, University of Oxford, Woodstock Road, Oxford, UK

Received  June 13, 2018 Published  June 2020

We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter. We also investigate a simple control problem in this context.
Keywords: Filtering, Optimal control, Robustness, Nonlinear expectation.
Mathematics Subject Classification: 62M20;60G35;93E11.
Citation: Samuel N. Cohen. Uncertainty and filtering of hidden Markov models in discrete time. Probability, Uncertainty and Quantitative Risk, 2020, 5 (0) : 4-. doi: 10.1186/s41546-020-00046-x
References:
[1]

Allan, A.L. and S.N. Cohen. (2019a). Parameter uncertainty in the Kalman–Bucy filter, SIAM J. Control Optim. 57, no. 3, 1646–1671. Google Scholar

[2]

Allan, A.L. and S.N. Cohen. (2020). Pathwise Stochastic Control with Applications to Robust Filtering, Ann. Appl. Prob. arXiv::1902.05434. Google Scholar

[3]

Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). Coherent measures of risk, Math. Finan. 9, no. 3, 203–228. Google Scholar

[4]

Başar, T. and P. Bernhard. (1991). H∞-Optimal Control and Related Minimax Design Problems, A Dynamic Game Approach, Birkhäuser, Basel. Google Scholar

[5]

Bain, A. and D. Crisan. (2009). Fundamentals of Stochastic Filtering, Springer, Berlin–Heidelberg–New York. Google Scholar

[6]

Bielecki, T.R., T. Chen, and I. Cialenco. (2017). Recursive construction of confidence regions, Electron. J. Stat. 11, no. 2, 4674–4700. Google Scholar

[7]

Boel, R.K., M.R. James, and I.R. Petersen. (2002). Robustness and risk-sensitive filtering, IEEE Trans. Autom. Control 47, no. 3, 451–461. Google Scholar

[8]

Cohen, S.N. and R.J. Elliott. (2010). A general theory of finite state backward stochastic difference equations, Stoch. Process. Appl. 120, no. 4, 442–466. Google Scholar

[9]

Cohen, S.N. and R.J. Elliott. (2011). Backward stochastic difference equations and nearly-time-consistent nonlinear expectations, SIAM J. Control Optim. 49, no. 1, 125–139. Google Scholar

[10]

Cohen, S.N. and R.J. Elliott. (2015). Stochastic Calculus and Applications, 2nd ed., Birkhäuser, New York. Google Scholar

[11]

Cohen, S.N. (2017). Data-driven nonlinear expectations for statistical uncertainty in decisions, Electron. J. Stat. 11, no. 1, 1858–1889. Google Scholar

[12]

Delbaen, F., S. Peng, and E. Rosazza Gianin. (2010). Representation of the penalty term of dynamic concave utilities, Finan. Stochast. 14, no. 3, 449–472. Google Scholar

[13]

Dey, S. and J.B. Moore. (1995). Risk-sensitive filtering and smoothing for hidden Markov models, Syst. Control Lett. 25, 361–366. Google Scholar

[14]

Douc, R., E. Moulines, J. Olsson, and R. van Handel. (2011). Consistency of the maximum likelihood estimator for general hidden Markov models, Ann. Stat. 39, no. 1, 474–513. Google Scholar

[15]

Duffie, D. and L.G. Epstein. (1992). Asset pricing with stochastic differential utility, Rev. Finan. Stud. 5, no. 3, 411–436. Google Scholar

[16]

El Karoui, N., S. Peng, and M.C. Quenez. (1997). Backward stochastic differential equations in finance, Math. Finan. 7, no. 1, 1–71. Google Scholar

[17]

Epstein, L.G. and M. Schneider. (2003). Recursive multiple-priors, J. Econ. Theory 113, 1–31. Google Scholar

[18]

Fagin, R. and J. Halpern. (1990). A new approach to updating beliefs, AUAI Press, Corvallis. Google Scholar

[19]

Föllmer, H. and A. Schied. (2002a). Convex measures of risk and trading constraints, Finan. Stochast. 6, 429–447. Google Scholar

[20]

Föllmer, H. and A. Schied. (2002b). Stochastic Finance: An Introduction in Discrete Time. Studies in Mathematics 27, de Gruyter, Berlin-New York. Google Scholar

[21]

Frittelli, M. and E. Rosazza Gianin. (2002). Putting order in risk measures, J. Bank. Financ. 26, no. 7, 1473–1486. Google Scholar

[22]

Graf, S. (1980). A Radon–Nikodym theorem for capacities, J. für die reine und angewandte Mathematik 320, 192–214. Google Scholar

[23]

Grimble, M.J. and A. El Sayed. (1990). Solution of the H∞ optimal linear filtering problem for discretetime systems, Trans. Acoust. Speech Sig. Process. IEEE 38, no. 7. Google Scholar

[24]

Hansen, L.P. and T.J. Sargent. (2005). Robust estimation and control under commitment, J. Econ. Theory 124, 258–301. Google Scholar

[25]

Hansen, L.P. and T.J. Sargent. (2007). Recursive robust estimation and control without commitment, J. Econ. Theory 136, no. 1, 1–27. Google Scholar

[26]

Hansen, L.P. and T.J. Sargent. (2008). Robustness, Princeton University Press, Princeton. Google Scholar

[27]

Huber, P.J. and E.M. Roncetti. (2009). Robust Statistics, 2nd edn., Wiley, Hoboken. Google Scholar

[28]

James, M.R., J.S. Baras, and R.J. Elliott. (1994). Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems, Trans. Autom. Control IEEE 39, no. 4, 780–792. https://doi.org/10.1109/9.286253. Google Scholar

[29]

Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, J. Basic Eng. ASME 82, 33–45. Google Scholar

[30]

Kalman, R.E. and R.S. Bucy. (1961). New results in linear filtering and prediction theory, J. Basic Eng. ASME 83, 95–108. Google Scholar

[31]

Keynes, J.M. (1921). A Treatise on Probability, Macmillan and Co., New York. Reprint BN Publishing, 2008. Google Scholar

[32]

Knight, F.H. (1921). Risk, Uncertainty and Profit, Houghton Mifflin, Boston. reprint Dover 2006. Google Scholar

[33]

Kupper, M. and W. Schachermayer. (2009). Representation results for law invariant time consistent functions, Math. Financ. Econ. 2, no. 3, 189–210. Google Scholar

[34]

Leroux, B.G. (1992). Maximum-likelihood estimation for hidden Markov models, Stoch. Process. Appl. 40, 127–143. Google Scholar

[35]

Peng, S. (2010). Nonlinear Expectations and Stochastic Calculus under Uncertainty, arxiv::1002.4546v1. Google Scholar

[36]

Riedel, F. (2004). Dynamic coherent risk measures, Stochast. Process. Appl. 112, no. 2, 185–200. Google Scholar

[37]

Rockafellar, R.T., S. Uryasev, and M. Zabarankin. (2006). Generalized deviations in risk analysis, Finan. Stochast. 10, 51–74. Google Scholar

[38]

Wald, A. (1945). Statistical decision functions which minimize the maximum risk, Ann. Math. 46, no. 2, 265–280. Google Scholar

[39]

Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London. Google Scholar

[40]

Wonham, W.N. (1965). Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. Control 2, 347–369. Google Scholar

[41]

Zhang, J., Y. Xia, and P. Shi. (2009). Parameter-dependent robust H∞ filtering for uncertain discrete-time systems, Automatica 45, 560–565. Google Scholar

show all references

References:
[1]

Allan, A.L. and S.N. Cohen. (2019a). Parameter uncertainty in the Kalman–Bucy filter, SIAM J. Control Optim. 57, no. 3, 1646–1671. Google Scholar

[2]

Allan, A.L. and S.N. Cohen. (2020). Pathwise Stochastic Control with Applications to Robust Filtering, Ann. Appl. Prob. arXiv::1902.05434. Google Scholar

[3]

Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). Coherent measures of risk, Math. Finan. 9, no. 3, 203–228. Google Scholar

[4]

Başar, T. and P. Bernhard. (1991). H∞-Optimal Control and Related Minimax Design Problems, A Dynamic Game Approach, Birkhäuser, Basel. Google Scholar

[5]

Bain, A. and D. Crisan. (2009). Fundamentals of Stochastic Filtering, Springer, Berlin–Heidelberg–New York. Google Scholar

[6]

Bielecki, T.R., T. Chen, and I. Cialenco. (2017). Recursive construction of confidence regions, Electron. J. Stat. 11, no. 2, 4674–4700. Google Scholar

[7]

Boel, R.K., M.R. James, and I.R. Petersen. (2002). Robustness and risk-sensitive filtering, IEEE Trans. Autom. Control 47, no. 3, 451–461. Google Scholar

[8]

Cohen, S.N. and R.J. Elliott. (2010). A general theory of finite state backward stochastic difference equations, Stoch. Process. Appl. 120, no. 4, 442–466. Google Scholar

[9]

Cohen, S.N. and R.J. Elliott. (2011). Backward stochastic difference equations and nearly-time-consistent nonlinear expectations, SIAM J. Control Optim. 49, no. 1, 125–139. Google Scholar

[10]

Cohen, S.N. and R.J. Elliott. (2015). Stochastic Calculus and Applications, 2nd ed., Birkhäuser, New York. Google Scholar

[11]

Cohen, S.N. (2017). Data-driven nonlinear expectations for statistical uncertainty in decisions, Electron. J. Stat. 11, no. 1, 1858–1889. Google Scholar

[12]

Delbaen, F., S. Peng, and E. Rosazza Gianin. (2010). Representation of the penalty term of dynamic concave utilities, Finan. Stochast. 14, no. 3, 449–472. Google Scholar

[13]

Dey, S. and J.B. Moore. (1995). Risk-sensitive filtering and smoothing for hidden Markov models, Syst. Control Lett. 25, 361–366. Google Scholar

[14]

Douc, R., E. Moulines, J. Olsson, and R. van Handel. (2011). Consistency of the maximum likelihood estimator for general hidden Markov models, Ann. Stat. 39, no. 1, 474–513. Google Scholar

[15]

Duffie, D. and L.G. Epstein. (1992). Asset pricing with stochastic differential utility, Rev. Finan. Stud. 5, no. 3, 411–436. Google Scholar

[16]

El Karoui, N., S. Peng, and M.C. Quenez. (1997). Backward stochastic differential equations in finance, Math. Finan. 7, no. 1, 1–71. Google Scholar

[17]

Epstein, L.G. and M. Schneider. (2003). Recursive multiple-priors, J. Econ. Theory 113, 1–31. Google Scholar

[18]

Fagin, R. and J. Halpern. (1990). A new approach to updating beliefs, AUAI Press, Corvallis. Google Scholar

[19]

Föllmer, H. and A. Schied. (2002a). Convex measures of risk and trading constraints, Finan. Stochast. 6, 429–447. Google Scholar

[20]

Föllmer, H. and A. Schied. (2002b). Stochastic Finance: An Introduction in Discrete Time. Studies in Mathematics 27, de Gruyter, Berlin-New York. Google Scholar

[21]

Frittelli, M. and E. Rosazza Gianin. (2002). Putting order in risk measures, J. Bank. Financ. 26, no. 7, 1473–1486. Google Scholar

[22]

Graf, S. (1980). A Radon–Nikodym theorem for capacities, J. für die reine und angewandte Mathematik 320, 192–214. Google Scholar

[23]

Grimble, M.J. and A. El Sayed. (1990). Solution of the H∞ optimal linear filtering problem for discretetime systems, Trans. Acoust. Speech Sig. Process. IEEE 38, no. 7. Google Scholar

[24]

Hansen, L.P. and T.J. Sargent. (2005). Robust estimation and control under commitment, J. Econ. Theory 124, 258–301. Google Scholar

[25]

Hansen, L.P. and T.J. Sargent. (2007). Recursive robust estimation and control without commitment, J. Econ. Theory 136, no. 1, 1–27. Google Scholar

[26]

Hansen, L.P. and T.J. Sargent. (2008). Robustness, Princeton University Press, Princeton. Google Scholar

[27]

Huber, P.J. and E.M. Roncetti. (2009). Robust Statistics, 2nd edn., Wiley, Hoboken. Google Scholar

[28]

James, M.R., J.S. Baras, and R.J. Elliott. (1994). Risk-sensitive control and dynamic games for partially observed discrete-time nonlinear systems, Trans. Autom. Control IEEE 39, no. 4, 780–792. https://doi.org/10.1109/9.286253. Google Scholar

[29]

Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, J. Basic Eng. ASME 82, 33–45. Google Scholar

[30]

Kalman, R.E. and R.S. Bucy. (1961). New results in linear filtering and prediction theory, J. Basic Eng. ASME 83, 95–108. Google Scholar

[31]

Keynes, J.M. (1921). A Treatise on Probability, Macmillan and Co., New York. Reprint BN Publishing, 2008. Google Scholar

[32]

Knight, F.H. (1921). Risk, Uncertainty and Profit, Houghton Mifflin, Boston. reprint Dover 2006. Google Scholar

[33]

Kupper, M. and W. Schachermayer. (2009). Representation results for law invariant time consistent functions, Math. Financ. Econ. 2, no. 3, 189–210. Google Scholar

[34]

Leroux, B.G. (1992). Maximum-likelihood estimation for hidden Markov models, Stoch. Process. Appl. 40, 127–143. Google Scholar

[35]

Peng, S. (2010). Nonlinear Expectations and Stochastic Calculus under Uncertainty, arxiv::1002.4546v1. Google Scholar

[36]

Riedel, F. (2004). Dynamic coherent risk measures, Stochast. Process. Appl. 112, no. 2, 185–200. Google Scholar

[37]

Rockafellar, R.T., S. Uryasev, and M. Zabarankin. (2006). Generalized deviations in risk analysis, Finan. Stochast. 10, 51–74. Google Scholar

[38]

Wald, A. (1945). Statistical decision functions which minimize the maximum risk, Ann. Math. 46, no. 2, 265–280. Google Scholar

[39]

Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities, Chapman and Hall, London. Google Scholar

[40]

Wonham, W.N. (1965). Some applications of stochastic differential equations to optimal nonlinear filtering, SIAM J. Control 2, 347–369. Google Scholar

[41]

Zhang, J., Y. Xia, and P. Shi. (2009). Parameter-dependent robust H∞ filtering for uncertain discrete-time systems, Automatica 45, 560–565. Google Scholar

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