Bayesian sequential testing with expectation constraints

Ankirchner, Stefan; Klein, Maike

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 51.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

We study a stopping problem arising from a sequential testing of two simple hypotheses H₀ and H₁ on the drift rate of a Brownian motion. We impose an expectation constraint on the stopping rules allowed and show that an optimal stopping rule satisfying the constraint can be found among the rules of the following type: stop if the posterior probability for H₁ attains a given lower or upper barrier; or stop if the posterior probability comes back to a fixed intermediate point after a sufficiently large excursion. Stopping at the intermediate point means that the testing is abandoned without accepting H₀ or H₁. In contrast to the unconstrained case, optimal stopping rules, in general, cannot be found among interval exit times. Thus, optimal stopping rules in the constrained case qualitatively differ from optimal rules in the unconstrained case.

Classification : 62L10, 60G40, 62L15
Mots-clés : Bayesian sequential testing, optimal stopping, expectation constraint

[1] S. Ankirchner, N. Kazi-Tani, M. Klein and T. Kruse, Stopping with expectation constraints: 3 points suffice. Electr. J. Probab. 24 (2019) 1–16. | MR | Zbl

[2] S. Ankirchner, M. Klein and T. Kruse, A verification theorem for optimal stopping problems with expectation constraints. Appl. Math. Optim. 79 (2019) 145–177. | DOI | MR | Zbl

[3] S. Ankirchner, M. Klein, T. Kruse and M. Urusov, On a certain local martingale in a general diffusion setting. Preprint (2018). | HAL

[4] E. Bayraktar and S. Yao, Dynamic Programming Principles for Optimal Stopping with Expectation Constraint. Preprint (2017). | arXiv

[5] L.I. Galtchouk, Optimality of the Wald SPRT for Processes with Continuous Time Parameter, in mODa 6 – Advances in Model-Oriented Design and Analysis, edited by A.C. Atkinson, P. Hackl and W.G. Müller. Physica-Verlag Heidelberg, HD (2001) 97–110. | MR

[6] D.P. Kennedy, On a constrained optimal stopping problem. J. Appl. Probab. 19 (1982) 631–641. | DOI | MR | Zbl

[7] R.S. Liptser and A.N. Shiryaev, Statistics of Random Processes: I. General Theory, Vol. 5 of Applications of Mathematics (New York). Springer-Verlag, Berlin, expandededition (2001). | MR | Zbl

[8] R.S. Liptser and A.N. Shiryaev, Statistics of Random Processes: II. Applications, Vol. 6 of Applications of Mathematics (New York). Springer-Verlag, Berlin, expandededition (2001). | MR | Zbl

[9] D.I. Lisovskii, Статистическиюе защачи щля некоторюх прощессов щиффузионного типа [Statistical problems for some diffusion-type processes]. Ph.D.thesis (2018).

[10] D.I. Lisovskii and A. Shiryaev, Sequential testing of two hypotheses for a stationary Ornstein–Uhlenbeck process. Theory Probab. Appl. 63 (2019) 580–593. | DOI | MR | Zbl

[11] G. Peskir and A. Shiryaev, Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2006). | MR | Zbl

[12] A.N. Shiryaev, Optimal Stopping Rules, Vol. 8 of Stochastic Modelling and Applied Probability. Translated fromthe 1976 Russian second edition by A.B. Aries, reprint of the 1978 translation. Springer-Verlag, Berlin (2008). | MR | Zbl

[13] A. Wald, Sequential Analysis. John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London (1947). | MR | Zbl

[14] A. Yashin, On a problem of sequential hypothesis testing. Theory Probab. Appl. 28 (1984) 157–165. | DOI | MR | Zbl