In this paper we investigate a class of swing options with firm constraints in view of the modeling of supply agreements. We show, for a fully general payoff process, that the premium, solution to a stochastic control problem, is concave and piecewise affine as a function of the global...

In continuous parametrized models with i.i.d. observations we consider finite quantizations. We study asymptotic properties of the estimators minimizing disparity between the observed and expected frequencies in the quantization cells, and asymptotic properties of the goodness of fit tests...

Necessary and sufficient conditions are given for the existence of optimal k-dimensional quantizers that minimize a distortion measure E{W(X)C(X - Q(X))}. An example is given in which a globally optimal quantizer does not exist.