Abstract
We are mainly interested in devising systematic approaches to the discretization of such problems in order to solve them numerically with the help of a computer.
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Notes
- 1.
This is likely to determine the complexity of the discretized optimization problem we seek to formulate.
- 2.
The notation
was introduced in Definition 3.30. - 3.
Actually, Pennanen considers non anticipativity constraints only.
- 4.
The finite set \(\mathbb {W}_{N}\) generally results from some sampling of the noise
or alternative quadrature methods. But, as recognized by Pennanen himself, in order to define a consistent discretization scheme, it is not enough to introduce the finite set \(\mathbb {W}_{N}\), but it is also necessary to define how the whole original set \(\mathbb {W}\) is mapped onto that finite set: this is why 
must be defined. - 5.
The following explanation can then be easily extended by considering
and 
for any intermediate \(t<T\). The truncation operator (which retains only the prefix of the process up to \(t\)) stands for the observation function \(h\) of the general theory. - 6.
Considering two scalar random variables with uniform distributions over bounded intervals, it can be checked that for the same number of cells, a pavement of a large surface in the plane with hexagons is more efficient in terms of the criterion (6.3) than a pavement with squares. The former cannot obviously be obtained as the Cartesian product of two one-dimensional quantizations.
- 7.
Other references dealing with scenario tree generation will be mentioned at Sect. 7.4.1.
- 8.
Optimal quantization (see Sect. 6.1.2) may be used to that purpose and many authors proposed various techniques to build up such trees—see e.g. [12, 61, 71, 114].
- 9.
For the notation \(\mathrm {O}\), see footnote 3 in Chap. 2.
- 10.
The authors are indebted to Prof. Benjamin Jourdain for preliminary results in this direction.
was introduced in Definition 
or alternative quadrature methods. But, as recognized by Pennanen himself, in order to define a consistent discretization scheme, it is not enough to introduce the finite set 
must be defined.
and 
for any intermediate Author information
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Carpentier, P., Chancelier, JP., Cohen, G., De Lara, M. (2015). Discretization Methodology for Problems with Static Information Structure (SIS). In: Stochastic Multi-Stage Optimization. Probability Theory and Stochastic Modelling, vol 75. Springer, Cham. https://doi.org/10.1007/978-3-319-18138-7_6
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