Sonlu Ufukta Genelleştirilmiş Kesinlik Eşdeğerlerinin Optimizasyonu

Öz

Bu makale, jenerik kesinlik eşdeğerlerinin optimizasyonundaki bazı açık sorunları ele almaktadır. Bu tür eşdeğerler, sistemin rasgele dinamiklerini modelleyen genel durum uzaylarında altta yatan kontrollü Markov zincirinin yollarında tanımlanan aşama başına sınırsız-üstü maliyet veya ödül fonksiyonlarının iskonto edilmiş toplamlarının artan fonksiyonelleri kullanılarak modellenmiştir. Bu tür işlevselliklere örnek olarak logaritmik ve güç araçlarının yanı sıra diğerleri arasında sağlam Riske Duyarlı tercihler verilebilir. Elde edilen kritik sonuçlar, her ikisi de gece ufku kurulumunda bir w-büyüme (dolayısıyla sınırsız) koşulunu sağlayan genel sınırsız-aşama üstü maliyet minimizasyonu ve aşama başına ödül maksimizasyonu için bu problemin çözümleriydi. Bu süreçte, dinamik programlama işleçlerinin önemsiz olmayan belirli kapatma özelliklerini oluşturuyoruz. Ek olarak, Portföy Tüketiminden gerçek hayattan bir örnek sunuyoruz.

Anahtar Kelimeler

• Kesinlik eşdeğerleri
• Tam Yol Birikmiş Kurulum
• Sınırsız ödüller ve maliyetler
• İndirimli yol toplamları
• Sağlam veya riske duyarlı terçihler

Optimization of Generalized Certainty Equivalents on the Finite Horizon

Abstract

This paper addresses some open issues in optimization of generic certainty equivalents. Such equivalents have been modelled using increasing functionals of the discounted sums of the per-stage unbounded-above cost or reward functions defined on the paths of the underlying controlled Markov chain on general state spaces which models the random dynamics of the system. Examples of such functionals include logarithmic and power utilities as well as the robust Risk-Sensitive preferences among others. The critical results that were obtained were the solutions of this problem for generic unbounded-above per-stage cost minimization and for per stage reward maximization, both satisfying a w-growth (hence unbounded) condition in the nite horizon setup. In the process, we establish certain nontrivial closure properties of the dynamic programming operators. In addition, we provide a real-life example from Portfolio Consumption.

Keywords

• Certainty equivalents
• Full-path accummulated setup
• Unbounded rewards and costs
• Discounted path sums
• Robust or risk-sensitive preferences

Kaynakça

1. Aliprantis C. D., Border K. C., Innite Dimensional Analysis: A Hitchhiker's Guide, Third Ed., Springer-Verlag, Berlin, 2006.
2. Altman E., Hordijk A., Spieksma F, M., Contraction Conditions for Average and α-discount Optimality in Countable State Markov Games with Unbounded Rewards, Math. Oper. Res. 22(3), 588-618, 1997.
3. Anderson E. W., Hansen L. P., Sargent T. J., Small Noise Methods for Risk Sensitive/Robust Economies, J. Econ. Dyn. Control 36, 468-500, 2012.
4. Basu A., Lontzek T., Schmedders, K., Zhao Y., The Social Cost of Carbon when we wish for Robustness, Accepted for publication in Management Science, 2022.
5. Bäuerle N., Rieder U., More Risk-Sensitive Markov Decision Processes, Math. Oper. Res. 39(1), 105-120, 2014.
6. Bäuerle N., Jasckiewicz A., Stochastic Optimal Growth Model with Risk-Sensitive Preferences. J. Eco, Theory 173, 181-200, 2018.
7. Bellman R., Dynamic Programming, Princeton University Press, Princeton, New Jersey, U.S.A, 1957.
8. Bertsekas D. P., Shreve, S. E., Stochastic Optimal Control: The Discrete-Time Case, Athena Scientic, Belmont, Massachusetts, 1996.

9. Bielecki T., Pliska S. R., Economic Properties of the Risk-Sensitive Criterion for Portfolio Management, Rev. Account. Fin. 2(2), 3-17, 2003.
10. Bommier A., Life-Cycle Preferences Revisited, J. Eur. Econ. Assoc. 11(6), 1290-1319, 2013.
11. Bommier, Lanz A. B., Zuber S., Models-as-usual for Unusual Risks? On the Value of Catastrophic Climate Change, J. Environ. Econ. Manag. 74, 1-22, 2015.
12. Bommier A., Kochov A., Le Grand F., On Monotone Recursive Preferences, Econometrica 85(5), 1433-1466, 2017.
13. Bommier, A. and F. Le Grand., Risk Aversion and Precautionary Savings in Dynamic Settings, Management Science 65(3), 1386-1397, 2019.
14. Borkar V. S., Probability Theory: An Advance Course, Springer-Verlag, New York,1995.
15. Cai Y., Lontzek T. S., The Social Cost of Carbon with Economic and Climate Risks, J. Pol. Econ. 127(6): 2684-2734, 2019.
16. Di Masi G. V., Stettner L., Risk-sensitive Control of Discrete-time Markov Processes with Infinite Horizon, SIAM J. Control Optim., 38(1): 61-78, 1999.
17. Epstein L. G., Zin S. E., Substitution, Risk Aversion, and the Temporal Behaviour of Consumption and Asset Returns: A Theoretical Framework, Econometrica 57(4): 937-969, 1989.
18. Epstein L. G., Zin S. E., Substitution, Risk Aversion, and the Temporal Behaviour of Consumption and Asset Returns: An Empirical Analysis, J. Pol. Econ. 99(2):263-286, 1991
19. Feinberg E. A., Kasyanov P. O., Zadoianchuk N. V., Average Cost Markov Decision Processes with Weakly Continuous Transition Probabilities, Math. Oper. Res. 37(4), 591-607. 2012
20. Feinberg E. A., Kasyanov P. O., Liang Y., Fatou's Lemma in Its Classical Form and Lebesque's Convergence Theorems for Varying Measures with Applications to Markov Decision Processes, Theory Probab. Appl. 65(2), 270-291, 2020.
21. Hansen L. P., Sargent T. J., Discounted Linear Exponential Quadratic Gaussian Control, IEEE Trans. on Auto. Control 40(5): 968-971, 1995.
22. Hansen L. P., Sargent T. J., Turmuhambetova G., Williams N., Robust Controland Model Misspecication, J. Econ. Theory 128, 45-90, 2006.
23. Hansen L. P., Sargent T. J., Recursive Robust Estimation and Control without Commitment, J. Econ. Theory 136(1): 1-27, 2007.
24. Hansen L. P., Sargent T. J., Robustness. Princeton Univ. Press, 2008.
25. Hinderer K., Foundations of Nonstationary Dynamic Programming with Discrete Time Parameter, Springer-Verlag, Berlin, 1970.
26. Howard R. A., Matheson J. E., Risk-sensitive Markov Decision Processes, Management Sc. 18(7), 356-369, 1972.
27. Jacobson D. H., Optimal Stochastic Linear Systems with Exponential Performance Criteria and Their Relation to Deterministic Dierential Games, IEEE Trans. Aut. Control, AC-18(2): 124-131, 1973.
28. Kreps D. M., Decision Problems with Expected Utility Criteria, I: Upper and Lower Convergent Utility, Math. Oper. Res. 2(1), 45-53, 1977.
29. Kreps D. M., Decision Problems with Expected Utility Criteria, II: Stationarity Math. Oper. Res. 2(3), 266-274, 1977.
30. Kreps D. M., Porteus E. L., Temporal Resolution of Uncertainty and Dynamic Choice Theory, Econometrica, 46(1): 185-200, 1978.
31. Kreps D. M., Porteus E. L., Dynamic Choice Theory and Dynamic Programming, Econometrica, 47(1): 91-100, 1979.
32. Meyn S. P., Tweedie R. L., Markov Chains and Stochastic Stability, Second Ed., Cambridge University Press, 2009.
33. Muliere P., Parmigiani G., Utility and Means in the 1930s, Statistic. Sc. 8(4), 421-432, 1993.
34. Nowak A. S., Notes on risk-sensitive Nash equilibria, in Advances in Dynamics, 95-109, Ann. Internatl. Soc. Dyn. Games, 7, Birkhauser, Boston, 2005.
35. Nowak A. S., Altman E., ε-Equilibria for Stochastic Games with Uncountable State Space and Unbounded Costs, SIAM J. Control Optim. 40(6), 1821-1839, 2002.
36. Tallarini Jr. T. D., Risk-Sensitive Business Cycles, J. Mon. Econ. 45, 507-532, 2000.
37. Whittle P., Risk-Sensitive Optimal Control, Wiley, Chichester, England, and New York, 1990.