This thesis consists of four papers treating the maximum principle for stochastic control problems. In the first paper we study the optimal control of a class of stochastic differential equations (SDEs) of mean-field type, where the coefficients are allowed to depend on the law of the process. Moreover, the cost functional of the control problem may also depend on the law of the process. Necessary and sufficient conditions for optimality are derived in the form of a maximum principle, which is also applied to solve the mean-variance portfolio problem. In the second paper, we study the problem of controlling a linear SDE where the coefficients are random and not necessarily bounded. We consider relaxed control processes, i.e. the control is defined as a process taking values in the space of probability measures on the control set. The main motivation is a bond portfolio optimization problem. The relaxed control processes are then interpreted as the portfolio weights corresponding to different maturity times of the bonds. We establish existence of an optimal control and necessary conditons for optimality in the form of a maximum principle, extended to include the family of relaxed controls. The third paper generalizes the second one by adding a singular control process to the SDE. That is, the control is singular with respect to the Lebesgue measure and its influence on the state is thus not continuous in time. In terms of the portfolio problem, this allows us to consider two investment possibilities - bonds (with a continuum of maturities) and stocks - and incur transaction costs between the two accounts. In the fourth paper we consider a general singular control problem. The absolutely continuous part of the control is relaxed in the classical way, i.e. the generator of the corresponding martingale problem is integrated with respect to a probability measure, guaranteeing the existence of an optimal control. This is shown to correspond to an SDE driven by a continuous orthogonal martingale measure. A maximum principle which describes necessary conditions for optimal relaxed singular control is derived. / QC 20100618

This thesis consists of four papers. In the first paper, Monte Carlo simulation for tail probabilities of heavy-tailed random walks is considered. Importance sampling algorithms are constructed by using mixtures of the original distribution with some other state-dependent distributions. Sufficient conditions under which the relative error of such algorithms is bounded are found, and the bound is calculated. A new mixture algorithm based on scaling of the original distribution is presented and compared to existing algorithms. In the second paper, Monte Carlo simulation of quantiles is treated. It is shown that by using importance sampling algorithms developed for tail probability estimation, efficient quantile estimators can be obtained. A functional limit of the quantile process under the importance sampling measure is found, and the variance of the limit process is calculated for regularly varying distributions. The procedure is also applied to the calculation of expected shortfall. The algorithms are illustrated numerically for a heavy-tailed random walk. In the third paper, large deviation probabilities for a sum of dependent random variables are derived. The dependence stems from a few underlying random variables, so-called factors. Each summand is composed of two parts: an idiosyncratic part and a part given by the factors. Conditions under which both factors and idiosyncratic components contribute to the large deviation behavior are found, and the resulting approximation is evaluated in a simple example. In the fourth paper, the asymptotic eigenvalue distribution of the exponentially weighted moving average covariance estimator is studied. Equations for the asymptotic spectral density and the boundaries of its support are found using the Marchenko-Pastur theorem. / QC 20100811

För att skatta skogsegenskaper utifrån data baserade på flygburen laserskanning används framförallt två olika metoder. Den första (areametoden) bygger uteslutande på höjden där laserpulserna reflekterats och en uppskattning av vilka pulser som reflekterats på marken och vilka som reflekterats i vegetationen. Den andra metoden (segmentmetoden) använder en segmentering av punkterna där laserpulserna reflekterats till enskilda träd. Detta är dock svårt att praktiskt genomföra, därför innehåller varje segment mellan 0 och cirka 10 träd. För att kunna utveckla och utvärdera resultaten insamlas även exakt information från ett antal provytor genom fältstudier. I denna rapport föreslås en icke-parametrisk modell för att skatta antalet träd i ett segment med segmentens area och art som förklarande variabler. Modellen skattas med hjälp av alla segment som finns inom en provyta, även de som inte helt ligger inom provytan. Modellen valideras på tre olika sätt och det visar sig att antalet träd i princip skattas väntevärdesriktigt. Dessutom förbättras inte resultatet nämnvärt med trädart som förklarande variabel. / To estimate forest characteristics based on airborne laser scanning data, two methods are used. The first one is on a raster cell level and uses solely the height of where the laser pulse is reflected and whether it is reflected on the ground or not. The second one is on an individual tree level and uses segmentation of the reflection points into individual trees. However, since it is difficult to segment into individual trees, every segment contains between 0 and about 10 trees. To be able to develop and validate the results, exact information about the trees in different field plots is gathered. In this report a nonparametric model is suggested to predict the number of trees in a segment, with area and species of the segments as predictors. In the estimation of the model, all segments within the field plots are used, even those only partly within. The model is validated in three different ways and estimates the number of trees with very small bias. Also the predictor, species of the segments, does not improve the results much.

University of Macau / Faculty of Science and Technology / Department of Mathematics

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