We analyze three different methods that can approximate the expected shortfall of a financial portfolio in a nested simulation. In this simulation process, the outer simulation generates risk scenarios, and the inner simulation approximates the value of the financial portfolio under each risk scenario. The first method is the most standard one, and therefore we call it ’the standard Monte Carlo method’. This method uses the same amount of computational cost for each inner simulation. The second method adapts the computational cost of the inner simulation to the output of the outer simulation. Therefore, we call it ’the adaptive sampling method’. This technique has already been proven to work more efficiently than the standard Monte Carlo method. The third method is called ’the multilevel Monte Carlo method’ (MLMC) and is based on the technique that has been introduced by Giles. This method approximates the expected shortfall multiple times and with different levels of accuracy. These estimators are used to give an accurate overall approximation of the expected shortfall. To the best of our knowledge, it has never been examined how efficiently the expected shortfall can be approximated in this manner by the MLMC method. This thesis will thoroughly explain how each method can be applied in a nested simulation to approximate the expected shortfall. In addition, an analysis is given on how to ensure that each method is used as efficiently as possible in the nested simulation. We also perform a numerical experiment in which we simulate a simplified version of a financial portfolio using stochastic processes. With the help of the numerical experiment, we examine which method is most efficient to approximate the expected shortfall.

In this thesis, we investigate whether maximizing the coverage of taxis can be beneficial when the goal is to minimize the waiting times of the clients. When dispatching taxis, often only current requests are taken into account and not future ones. We examined how beneficial it can be to take coverage into account. For taxi companies it is important to keep their customers satisfied, by serving them as quickly as possible. Often companies assign the taxis to the nearest requests. We investigate whether there are better dispatch methods. We developed three dispatch policies which use ILP models, and discussed what the best option is. The first policy does not consider the coverage and minimizes the waiting times of the clients among the requests that are taking place at that moment. The second one aims to maintain a good coverage and short waiting times when assigning taxis to requests. The third one also aims to maintain a good coverage and short waiting times when assigning taxis to requests and also relocates taxis to gain a better covered area. Those policies can be a contribution for taxi companies because they help to serve clients more quickly. To determine what the best way is of dispatching taxis, we run a simulation on the different policies on a real-data map and compared the results. The literature over taxi dispatch mostly conducts research on a small area like a city or a village. In this thesis, taxis dispatch takes place on a larger area where taxis commute between cities.