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stochastic programming problems; probability constraints; stochastic dominance; stability; Wasserstein metric; ${\cal L}_{1}$ norm; Lipschitz property; empirical estimates; scenario; error approximation; financial applications; loan; debtor; installments; mortgage; bank
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 ${\cal L}_{1}$ 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.
[1] L.Dai, Chen, C. H., Birge, J. R.: Convergence properties of two-stage stochastic programming. J. Optim. Theory Appl. 106 (2000), 489-509. DOI 10.1023/a:1004649211111 | MR 1797371
[2] Dupačová, J., Wets, R. J.-B.: Asymptotic behaviour of statistical estimates and optimal solutions of stochastic optimization problems. Ann. Statist. 16 (1984), 1517-1549. DOI 10.1214/aos/1176351052 | MR 0964937
[3] Dupačová, J., Hurt, J., Štěpán, J.: Stochastic Modelling in Economics and Finance. Kluwer, Dordrecht 2002. MR 2008457
[4] Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 (1963), 301, 13-30. DOI 10.1080/01621459.1963.10500830 | MR 0144363
[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.: Probabilistic bounds (via large deviations) for the solutions of stochastic programming problems. Ann. Oper. Res. 56 (1995), 189-208. DOI 10.1007/bf02031707 | MR 1339792 | Zbl 0835.90055
[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. MR 0519478
[8] Kaňková, V.: An approximative solution of stochastic optimization problem. In: Trans. 8th. Prague Conference, Academia, Prague 1978, pp. 349-353. DOI 10.1007/978-94-009-9857-5_33 | MR 0536792
[9] Kaňková, V.: Uncertainty in stochastic programming. 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. DOI 10.1007/bfb0007116 | MR 0891003
[10] Kaňková, V.: On the stability in stochastic programming: the case of individual probability constraints. Kybernetika 33 (1997), 5, 525-546. MR 1603961 | Zbl 0908.90198
[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. Zbl 1162.90528
[12] Kaňková, V.: Multistage stochastic programs via autoregressive sequences and individual probability constraints. Kybernetika 44 (2008), 2, 151-170. MR 2428217
[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.: Thin and heavy tails in stochastic programming. Kybernetika 51 (2015), 3, 433-456. DOI 10.14736/kyb-2015-3-0433 | MR 3391678
[15] Kaňková, V.: Scenario generation via ${\cal 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.: Mathematical Formulas for Economists. Third edition. Springer Science and Media, 2006. DOI 10.1007/978-3-662-12431-4 | MR 1614635
[18] Pflug, G. Ch.: Scenarion tree generation for multiperiod finncial optimization by optimal discretizatin. Math. Program. Ser. B 89 (2001), 251-271. DOI 10.1007/pl00011398 | MR 1816503
[19] Pflug, G. Ch.: Stochastic Optimization and Statistical Inference. 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. DOI 10.1016/s0927-0507(03)10007-2 | MR 2051793
[20] Rockafellar, R., Wets, R. J. B.: Variational Analysis. Springer, Berlin 1983. DOI 10.1007/978-3-642-02431-3 | Zbl 0888.49001
[21] Römisch, W., Schulz, R.: Stability of solutions for stochastic programs with complete recourse. Math. Oper. Res. 18 (1993), 590-609. DOI 10.1287/moor.18.3.590 | MR 1250562
[22] Römisch, W.: Stability of Stochastic Programming Problems. 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. DOI 10.1016/s0927-0507(03)10008-4 | MR 2051791
[23] Salinetti, G., Wets, R. J. B.: On the convergence of closed-valued measurable multifunctions. Trans. Amer. Math. Society 266 (1981), 1, 275-289. DOI 10.1090/s0002-9947-1981-0613796-3 | MR 0613796
[24] Schulz, R.: Rates of convergence in stochastic programs with complete integer recourse. SIAM J. Optim. 6 (1996), 4, 1138-1152. DOI 10.1137/s1052623494271655 | MR 1416533
[25] Shapiro, A.: Quantitative stability in stochastic programming. Math. Program. 67 (1994), 99-108. DOI 10.1007/bf01582215 | MR 1300821 | Zbl 0828.90099
[26] Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming (Modeling and Theory). Published by Society for Industrial and Applied Mathematics and Mathematical Programming Society, Philadelphia 2009. DOI 10.1137/1.9780898718751 | MR 2562798 | Zbl 1302.90003
[27] Šmíd, M.: The expected loss in the discretization of multistage stochastic programming problems-estimation and convergence rate. Ann. Oper. Res. 165 (2009), 29-45. DOI 10.1007/s10479-008-0355-9 | MR 2470981
[28] Šmíd, M., Dufek, J.: Multi-period Factor Model of Loan Portfolio (July 10, 2016). DOI 10.2139/ssrn.2703884
[29] Shorack, G. R., Wellner, J. A.: Empirical Processes and Applications to Statistics. Wiley, New York 1986. DOI 10.1137/1.9780898719017 | MR 0838963
[30] Wets, R. J. B.: A Statistical Approach to the Solution of Stochastic Programs with (Convex) Simple Recourse. Research Report, University Kentucky 1974. MR 0727454
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