For citation:
Sidorov S. P., Zakharova E. A. On the Error of Approximation by Means of Scenario Trees with Depth 1. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 3, pp. 95-99. DOI: 10.18500/1816-9791-2013-13-3-95-99
On the Error of Approximation by Means of Scenario Trees with Depth 1
Let¤n denote the set of scenario trees with depth 1 and n scenarios. LetX = (0 · x1 < . . . < xn · 1) and let¤n(X) denote the set of all scenario trees of depth 1 with the scenarios X = (0 · x1 < . . . < xn · 1). Let G be a probability distribution defined on [0, 1] and H be a subset of measurable functions defined on [0, 1]. Let dH,X(G) = inf ˜G∈¤n(X) dH(G, ˜ G) and dH(G) = inf ˜G∈¤n dH(G, ˜ G), where dH(G, ˜ G) := suph∈H ¯¯¯ R h dG − R h d˜G ¯¯¯ . The main goal of the paper is to estimate dH(G,X) and dH(G) in the case when the set H is a subset of all algebraical polynomials of degree · 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.
- Hochreiter R., Pflug G. Ch. Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Annals of Operations Research, 2007, vol. 152, no. 1, pp. 257–272.
- Heitsch H., R¨omisch W. Scenario tree modeling for multistage stochastic programs. Math. Program., 2009, vol. 118, no. 2, pp. 371–406.
- Rockafellar R., Uryasev S. Optimization of Conditional Value-at-Risk. The Journal of Risk, 2000, vol. 2, no. 3, pp. 21–41.
- Dupacova J., Consigli G., Wallace S. W. Generating Scenarios for Multistage Stochastic Programs. Annals of Operations Research, 2000, vol. 100, pp. 25–53.
- Hoyland K., Wallace S. W. Generating Scenario Trees for Multistage Decision Problems. Management Science, 2001, vol. 47, pp. 295–307. 6. Traub J. F., Wo´zniakowski H. A general theory of optimal algorithms. New York, Academic Press, 1980. 341p.
- 1174 reads