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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.

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

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.

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[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.

[24]

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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[39]

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

[40]

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

[41]

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

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