Stability, empirical estimates and scenario generation in stochastic optimization - applications in finance

Vlasta Kaňková

Kybernetika (2017)

  • Volume: 53, Issue: 6, page 1026-1046
  • ISSN: 0023-5954

Abstract

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Economic and financial processes are mostly simultaneously influenced by a random factor and a decision parameter. While the random factor can be hardly influenced, the decision parameter can be usually determined by a deterministic optimization problem depending on a corresponding probability measure. However, in applications the “underlying” probability measure is often a little different, replaced by empirical one determined on the base of data or even (for numerical reason) replaced by simpler (mostly discrete) one. Consequently, real one and approximate one correspond to applications. In the paper we try to investigate their relationship. To this end we employ the results on stability based on the Wasserstein metric and ℒ 1 norm, their applications to empirical estimates and scenario generation. Moreover, we apply the achieved new results to simple financial applications. The corresponding model will a problem of stochastic programming.

How to cite

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Kaňková, Vlasta. "Stability, empirical estimates and scenario generation in stochastic optimization - applications in finance." Kybernetika 53.6 (2017): 1026-1046. <http://eudml.org/doc/294823>.

@article{Kaňková2017,
abstract = {Economic and financial processes are mostly simultaneously influenced by a random factor and a decision parameter. While the random factor can be hardly influenced, the decision parameter can be usually determined by a deterministic optimization problem depending on a corresponding probability measure. However, in applications the “underlying” probability measure is often a little different, replaced by empirical one determined on the base of data or even (for numerical reason) replaced by simpler (mostly discrete) one. Consequently, real one and approximate one correspond to applications. In the paper we try to investigate their relationship. To this end we employ the results on stability based on the Wasserstein metric and $\{\mathcal \{L\}\}_\{1\}$ norm, their applications to empirical estimates and scenario generation. Moreover, we apply the achieved new results to simple financial applications. The corresponding model will a problem of stochastic programming.},
author = {Kaňková, Vlasta},
journal = {Kybernetika},
keywords = {stochastic programming problems; probability constraints; stochastic dominance; stability; Wasserstein metric; $\{\mathcal \{L\}\}_\{1\}$ norm; Lipschitz property; empirical estimates; scenario; error approximation; financial applications; loan; debtor; installments; mortgage; bank},
language = {eng},
number = {6},
pages = {1026-1046},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Stability, empirical estimates and scenario generation in stochastic optimization - applications in finance},
url = {http://eudml.org/doc/294823},
volume = {53},
year = {2017},
}

TY - JOUR
AU - Kaňková, Vlasta
TI - Stability, empirical estimates and scenario generation in stochastic optimization - applications in finance
JO - Kybernetika
PY - 2017
PB - Institute of Information Theory and Automation AS CR
VL - 53
IS - 6
SP - 1026
EP - 1046
AB - Economic and financial processes are mostly simultaneously influenced by a random factor and a decision parameter. While the random factor can be hardly influenced, the decision parameter can be usually determined by a deterministic optimization problem depending on a corresponding probability measure. However, in applications the “underlying” probability measure is often a little different, replaced by empirical one determined on the base of data or even (for numerical reason) replaced by simpler (mostly discrete) one. Consequently, real one and approximate one correspond to applications. In the paper we try to investigate their relationship. To this end we employ the results on stability based on the Wasserstein metric and ${\mathcal {L}}_{1}$ norm, their applications to empirical estimates and scenario generation. Moreover, we apply the achieved new results to simple financial applications. The corresponding model will a problem of stochastic programming.
LA - eng
KW - stochastic programming problems; probability constraints; stochastic dominance; stability; Wasserstein metric; ${\mathcal {L}}_{1}$ norm; Lipschitz property; empirical estimates; scenario; error approximation; financial applications; loan; debtor; installments; mortgage; bank
UR - http://eudml.org/doc/294823
ER -

References

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  1. L.Dai, Chen, C. H., Birge, J. R., 10.1023/a:1004649211111, J. Optim. Theory Appl. 106 (2000), 489-509. MR1797371DOI10.1023/a:1004649211111
  2. Dupačová, J., Wets, R. J.-B., 10.1214/aos/1176351052, Ann. Statist. 16 (1984), 1517-1549. MR0964937DOI10.1214/aos/1176351052
  3. Dupačová, J., Hurt, J., Štěpán, J., Stochastic Modelling in Economics and Finance., Kluwer, Dordrecht 2002. MR2008457
  4. Hoeffding, W., 10.1080/01621459.1963.10500830, J. Amer. Statist. Assoc. 58 (1963), 301, 13-30. MR0144363DOI10.1080/01621459.1963.10500830
  5. Houda, M., Kaňková, V., Empirical estimates in economic and financial optimization problems., Bull. Czech Econometr. Soc. 19 (2012), 29, 50-69. 
  6. Kaniovski, Y. M., King, A. J., Wets, R. J.-B., 10.1007/bf02031707, Ann. Oper. Res. 56 (1995), 189-208. Zbl0835.90055MR1339792DOI10.1007/bf02031707
  7. Kaňková, V., Optimum solution of a stochastic optimization problem with unknown parameters., In: Trans. 7th. Prague Conf. 1974, Academia, Prague 1977, pp. 239-244. MR0519478
  8. Kaňková, V., 10.1007/978-94-009-9857-5_33, In: Trans. 8th. Prague Conference, Academia, Prague 1978, pp. 349-353. MR0536792DOI10.1007/978-94-009-9857-5_33
  9. Kaňková, V., 10.1007/bfb0007116, In: Proc. Inter. Conf. on Stoch. Optim., Kiev 1984 (V. I Arkin and R. J.-B. Wets, eds.), Lecture Notes in Control and Information Sciences 81, Springer, Berlin 1986, pp. 393-401. MR0891003DOI10.1007/bfb0007116
  10. Kaňková, V., On the stability in stochastic programming: the case of individual probability constraints., Kybernetika 33 (1997), 5, 525-546. Zbl0908.90198MR1603961
  11. Kaňková, V., Houda, M., Empirical estimates in stochastic programming., In: Proc. Prague Stochastics 2006 (M. Hušková and M. Janžura, eds.), MATFYZPRESS, Prague 2006, pp. 426-436. Zbl1162.90528
  12. Kaňková, V., Multistage stochastic programs via autoregressive sequences and individual probability constraints., Kybernetika 44 (2008), 2, 151-170. MR2428217
  13. Kaňková, V., Empirical estimates in optimization problems; survey with special regard to heavy tails and dependent samples., Bull.Czech Econometric. soc. 19 (2012), 30, 92-111. 
  14. Kaňková, V., Houda, M., 10.14736/kyb-2015-3-0433, Kybernetika 51 (2015), 3, 433-456. MR3391678DOI10.14736/kyb-2015-3-0433
  15. Kaňková, V., Scenario generation via ℒ 1 norm., In: Proc. 33rd Inter. Conf. Mathematical Methods in Economics 2015 (D. Marinčík, J. Ircingová and P. Janeček, eds.), Published by West Bohemia, Plzeň 2015, pp. 331-336. 
  16. Kaňková, V., A note on optimal value of loans., In: Proc. 34th Inter. Conf. Mathematical methods in economics 2016 (A. Kocourek and M. Vavroušek, eds.), Technical University Liberec, Liberec 2016, pp, 371-376. 
  17. Luderer, B., Nollau, V., Vetters, K., 10.1007/978-3-662-12431-4, Springer Science and Media, 2006. MR1614635DOI10.1007/978-3-662-12431-4
  18. Pflug, G. Ch., 10.1007/pl00011398, Math. Program. Ser. B 89 (2001), 251-271. MR1816503DOI10.1007/pl00011398
  19. Pflug, G. Ch., 10.1016/s0927-0507(03)10007-2, In: Stochastic Programming, Handbooks in Operations Research and Managemennt Science, Vol. 10 (A. Ruszczynski and A. A. Shapiro, eds.), Elsevier, Amsterdam 2003, pp. 427-480. MR2051793DOI10.1016/s0927-0507(03)10007-2
  20. Rockafellar, R., Wets, R. J. B., 10.1007/978-3-642-02431-3, Springer, Berlin 1983. Zbl0888.49001DOI10.1007/978-3-642-02431-3
  21. Römisch, W., Schulz, R., 10.1287/moor.18.3.590, Math. Oper. Res. 18 (1993), 590-609. MR1250562DOI10.1287/moor.18.3.590
  22. Römisch, W., 10.1016/s0927-0507(03)10008-4, In: Stochastic Programming, Handbooks in Operations Research and Managemennt Science, Vol. 10 (A. Ruszczynski and A. A. Shapiro, eds.), Elsevier, Amsterdam 2003, pp. 483-554. MR2051791DOI10.1016/s0927-0507(03)10008-4
  23. Salinetti, G., Wets, R. J. B., 10.1090/s0002-9947-1981-0613796-3, Trans. Amer. Math. Society 266 (1981), 1, 275-289. MR0613796DOI10.1090/s0002-9947-1981-0613796-3
  24. Schulz, R., 10.1137/s1052623494271655, SIAM J. Optim. 6 (1996), 4, 1138-1152. MR1416533DOI10.1137/s1052623494271655
  25. Shapiro, A., 10.1007/bf01582215, Math. Program. 67 (1994), 99-108. Zbl0828.90099MR1300821DOI10.1007/bf01582215
  26. Shapiro, A., Dentcheva, D., Ruszczynski, A., 10.1137/1.9780898718751, Published by Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia 2009. Zbl1302.90003MR2562798DOI10.1137/1.9780898718751
  27. Šmíd, M., 10.1007/s10479-008-0355-9, Ann. Oper. Res. 165 (2009), 29-45. MR2470981DOI10.1007/s10479-008-0355-9
  28. Šmíd, M., Dufek, J., 10.2139/ssrn.2703884 DOI10.2139/ssrn.2703884
  29. Shorack, G. R., Wellner, J. A., 10.1137/1.9780898719017, Wiley, New York 1986. MR0838963DOI10.1137/1.9780898719017
  30. Wets, R. J. B., A Statistical Approach to the Solution of Stochastic Programs with (Convex) Simple Recourse., Research Report, University Kentucky 1974. MR0727454

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