Geodesic
On the error of approximation by means of scenario trees with depth~1
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 3, pp. 95-99.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\Lambda_n$ denote the set of scenario trees with depth 1 and $n$ scenarios. Let $X=(0\le x_1\dots$ and let $\Lambda_n(X)$ denote the set of all scenario trees of depth 1 with the scenarios $X=(0\le x_1\dots$. Let $G$ be a probability distribution defined on $[0,1]$ and $H$ be a subset of measurable functions defined on $[0,1]$. Let $d_{H,X}(G)=\inf_{\tilde G\in\Lambda_n(X)}d_H(G,\tilde G)$ and $d_H(G)=\inf_{\tilde G\in\Lambda_n}d_H(G,\tilde G)$, where $d_H(G,\tilde G):=\sup_{h\in H}\left|\int h\,dG-\int h\,d\tilde G\right|$. The main goal of the paper is to estimate $d_H(G,X)$ and $d_H(G)$ in the case when the set $H$ is a subset of all algebraical polynomials of degree $\leq n$. Thus, the paper is examined the error of approximation of a continuous distribution $G$ by means of scenario trees with depth 1 and matching the first $n$ moments. 
@article{ISU_2013_13_3_a13,
     author = {E. A. Zakharova and S. P. Sidorov},
     title = {On the error of approximation by means of scenario trees with depth~1},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {95--99},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2013_13_3_a13/}
}
TY  - JOUR
AU  - E. A. Zakharova
AU  - S. P. Sidorov
TI  - On the error of approximation by means of scenario trees with depth~1
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2013
SP  - 95
EP  - 99
VL  - 13
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2013_13_3_a13/
LA  - ru
ID  - ISU_2013_13_3_a13
ER  - 
%0 Journal Article
%A E. A. Zakharova
%A S. P. Sidorov
%T On the error of approximation by means of scenario trees with depth~1
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2013
%P 95-99
%V 13
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2013_13_3_a13/
%G ru
%F ISU_2013_13_3_a13
E. A. Zakharova; S. P. Sidorov. On the error of approximation by means of scenario trees with depth~1. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 13 (2013) no. 3, pp. 95-99. http://geodesic.mathdoc.fr/item/ISU_2013_13_3_a13/

[1] Hochreiter R., Pflug G. Ch., “Financial scenario generation for stochastic multi-stage decision processes as facility location problems”, Annals of Operations Research, 152:1 (2007), 257–272 | DOI | MR | Zbl

[2] Heitsch H., Römisch W., “Scenario tree modeling for multistage stochastic programs”, Math. Program., 118:2 (2009), 371–406 | DOI | MR | Zbl

[3] Rockafellar R., Uryasev S., “Optimization of Conditional Value-at-Risk”, The Journal of Risk, 2:3 (2000), 21–41

[4] Dupacova J., Consigli G., Wallace S. W., “Generating Scenarios for Multistage Stochastic Programs”, Annals of Operations Research, 100 (2000), 25–53 | DOI | MR | Zbl

[5] Hoyland K., Wallace S. W., “Generating Scenario Trees for Multistage Decision Problems”, Management Science, 47 (2001), 295–307 | DOI | Zbl

[6] Traub J. F., Woźniakowski H., A general theory of optimal algorithms, Academic Press, New York, 1980, 341 pp. | MR | MR | Zbl