Kybernetika 54 no. 6, 1184-1200, 2018

Multistage multivariate nested distance: An empirical analysis

Sebastiano VitaliDOI: 10.14736/kyb-2018-6-1184

Abstract:

Multistage stochastic optimization requires the definition and the generation of a discrete stochastic tree that represents the evolution of the uncertain parameters in time and space. The dimension of the tree is the result of a trade-off between the adaptability to the original probability distribution and the computational tractability. Moreover, the discrete approximation of a continuous random variable is not unique. The concept of the best discrete approximation has been widely explored and many enhancements to adjust and fix a stochastic tree in order to represent as well as possible the real distribution have been proposed. Yet, often, the same generation algorithm can produce multiple trees to represent the random variable. Therefore, the recent literature investigates the concept of distance between trees which are candidate to be adopted as stochastic framework for the multistage model optimization. The contribution of this paper is to compute the nested distance between a large set of multistage and multivariate trees and, for a sample of basic financial problems, to empirically show the positive relation between the tree distance and the distance of the corresponding optimal solutions, and between the tree distance and the optimal objective values. Moreover, we compute a lower bound for the Lipschitz constant that bounds the optimal value distance.

Keywords:

multistage stochastic optimization, nested distance, portfolio models

Classification:

90C15, 60B05, 62P05

paper.pdf

References:

  1. J. R. Birge and F. Louveaux: Introduction to Stochastic Programming. Springer Science and Business Media, 2011.   CrossRef
  2. G. Consigli, V. Moriggia, E. Benincasa, G. Landoni, F. Petronio, S. Vitali, M. di Tria, M. Skoric and A. Uristani: Optimal multistage defined-benefit pension fund management. In: Recent Advances in Commmodity and Financial Modeling: Quantitative methods in Banking, Finance, Insurance, Energy and Commodity markets (G. Consigli, S. Stefani, and G. Zambruno eds.), Springer's International Series in Operations Research and Management Science, 2017.   DOI:10.1007/978-3-319-61320-8\_13
  3. J. Dupačová, J. Hurt and J. Štěpán: Stochastic Modeling in Economics and Finance. Applied Optimization, Springer, 2002.   DOI:10.1007/b101992
  4. S. Kilianová and G. C. Pflug: Optimal pension fund management under multi-period risk minimization. Ann. Oper. Res. 166 (2009), 1, 261-270.   DOI:10.1007/b101992
  5. M. Kopa and B. Petrová: Multistage risk premiums in portfolio optimization. Kybernetika 53 (2017), 6, 992-1011.   DOI:10.14736/kyb-2017-6-0992
  6. M. Kopa, V. Moriggia and S. Vitali: Individual optimal pension allocation under stochastic dominance constraints. Ann. Oper. Res. 260 (2018), 1,2, 255-291.   DOI:10.1007/s10479-016-2387-x
  7. R. M. Kovacevic and A. Pichler: Tree approximation for discrete time stochastic processes: a process distance approach. Ann. Oper. Res. 235 (2015), 1, 395-421.   DOI:10.1007/s10479-015-1994-2
  8. F. Maggioni and G. C. Pflug: Bounds and approximations for multistage stochastic programs. SIAM J. Optim. 26 (2016), 1, 831-855.   DOI:10.1137/140971889
  9. F. Maggioni, E. Allevi and M. Bertocchi: Bounds in multistage linear stochastic programming. J. Optim. Theory Appl. 163 (2014), 1, 200-229.   DOI:10.1007/s10957-013-0450-1
  10. F. Maggioni, E. Allevi and M. Bertocchi: Monotonic bounds in multistage mixed-integer liner stochastic programming. Comput. Management Sci. 13 (2016), 3, 423-457.   DOI:10.1007/s10287-016-0254-5
  11. G. C. Pflug and A. Pichler: A distance for multistage stochastic optimization models. SIAM J. Optim. 22 (2012), 1, 1-23.   DOI:10.1137/110825054
  12. G. C. Pflug and A. Pichler: Multistage Stochastic Optimization. Springer, 2014.   DOI:10.1007/978-3-319-08843-3
  13. G. C. Pflug and A. Pichler: Convergence of the smoothed empirical process in nested distance. Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut fűr Mathematik (J. L. Higle, W. Römisch, and S. Surrajeet, eds.), 2015.   CrossRef
  14. G. C. Pflug and A. Pichler: From empirical observations to tree models for stochastic optimization: Convergence properties. SIAM J. Optim. 26 (2016), 3, 1715-1740.   DOI:10.1137/15m1043376
  15. W. B. Powell: Clearing the jungle of stochastic optimization. Informs TutORials, 2014.   DOI:10.1287/educ.2014.0128
  16. T. R. Rockafellar and S. Uryasev: Optimization of conditional value-at-risk. J. Risk 2 (2000), 21-42.   DOI:10.21314/jor.2000.038
  17. A. Shapiro, D. Dentcheva and A. Ruszczyński: Lectures on stochastic programing. Modeling and Theory. SIAM Math. Programm. Soc. 2009.   CrossRef
  18. A. V. Timonina: Multi-stage stochastic optimization: the distance between stochastic scenario processes. Computat. Management Sci. 12 (2015), 1, 171-195.   DOI:10.1007/s10287-013-0185-3
  19. S. Vitali, V. Moriggia and M. Kopa: Optimal pension fund composition for an Italian private pension plan sponsor. Comput. Management Sci. 14 (2017), 1, 135-160.   DOI:10.1007/s10287-016-0263-4