Anmerkungen:
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Beschreibung:
The first part of this thesis deals with the sensitivity and statistical estimation of the optimal value of a Markov decision model (MDM) in the transition probability function, i.e.\ the family of all transition probabilities. Such models are used for modelling {\color{black}stochastic optimization problems with sequential decision making} which appear in many application areas. Since in practice, the used MDM is most often less complex than the underlying `true' MDM, we first discuss the impact of a reduction of the model complexity in the transition probability function on the optimal value of the MDM, i.e.\ the solution of the underlying stochastic control problem. Besides a statement on the continuity of the optimal value regarded as a real-valued functional on a set of transition probability functions, we will in particular introduce a sort of derivative of this functional which can be used to measure the (first-order) sensitivity of the optimal value w.r.t.\ deviations in the transition probability function. In addition, we perform a statistical analysis of the optimal value of a MDM where the underlying transition probability function is unknown, a situation that often occurs in practice. By limiting ourselves to a simple MDM in which the transition probability function is generated only by a single distribution function, we show that the optimal value construed as a real-valued functional defined on a set of distribution functions is continuous and functionally differentiable in a certain sense. By means of these regularity properties, we discuss the asymptotics of suitable estimators for the optimal value of the MDM in nonparametric and parametric statistical models. Our theoretical findings in the first part of this thesis are illustrated by means of optimization problems in inventory control and mathematical finance. The second part of this thesis is devoted to the nonparametric estimation of risk measures of collective risks in a non-homogeneous individual risk model in connection with the ...