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=== Distributional assumption ===
The formulation of the above two-stage problem assumes that the second-stage data <math>\xi</math> can be modeled as a random vector with a '''''known''''' probability distribution (not just uncertain). This would be justified in many situations. For example, <math>\xi</math> could be information derived from historical data and the distribution does not significantly change over the considered period of time. In such situations one may reliably estimate the required probability distribution and the optimization ''on average'' could be justified by the [[law of large numbers]]. Another example is that <math>\xi</math> could be realizations of a simulation model whose outputs are stochastic. The empirical distribution of the sample could be used as an approximation to the true but unknown output distribution.
=== Discretization ===
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</math></div>
By the [[Law of Large Numbers]] we have that, under some regularity conditions <math>\frac{1}{N} \sum_{j=1}^N Q(x,\xi^j)</math> converges pointwise with probability 1 to <math>E[Q(x,\xi)]</math> as <math>N \rightarrow \infty</math>. Moreover, under mild additional conditions the convergence is uniform. We also have <math>E[\hat{g}_N(x)]=g(x)</math>, i.e., <math>\hat{g}_N(x)</math> is an ''unbiased'' estimator of <math>g(x)</math>. Therefore, it is natural to expect that the optimal value and optimal solutions of the SAA problem converge to their counterparts of the true problem as <math>N \rightarrow \infty</math>.
===Consistency of SAA estimators===
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</math></div>
where <math>X_N</math> is a subset of <math>\mathbb{R}^n</math> depending on the sample and therefore is random. Nevertheless, consistency results for SAA estimators can still be derived under some additional assumptions:
# Suppose that there exists a compact set <math>C \subset \mathbb{R}^n</math> such that
#* the set <math>S</math> of optimal solutions of the true problem is nonempty and is contained in <math>C</math>
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==Economic applications==
Stochastic dynamic programming is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example; often it is used by resource economists to analyze [[Bioeconomics|bioeconomic problems]]{{
==Software tools==
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