<p>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.</p> / <p>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.</p>

This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R. The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied. Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied. The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.

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This thesis consists of four papers. In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities. In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes. In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric. In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.

The thesis treats three different areas; (i) Ranked increments of stable processes and ranked excursions of Brownian motion, (ii) Sufficient capital levels for banks, and (iii) Trading strategies for reduction of the fluctuations of revenues for power plants. The first part is a theoretcial investigation involved with the calculation of distribution functions concerning special properties of stable processes. The second part is a description of a framework in which the sufficiency of capital levels for banks can be evaluated. The third part is a typical example of how financial mathematics can be used to derive practical methods applicable in risk management of energy derivatives and real options. Altogether, five papers are presented.

An outstanding problem in physics is to find a unified framework for quantum mechanics and general relativity. This is required for a better understanding of black holes and the early cosmology of the universe. String theory provides such a unification. In this thesis, we study aspects of compactifications of type IIB string theory. In the first part of the thesis, we study four-dimensional black holes consisting of D3-branes wrapping cycles in the compact dimensions. We discuss the correspondence between these black holes, topological string theory and matrix models. We then study the influence of black holes on the stability of flux compactifications. In the second part of the thesis, we turn to investigations of the type IIB landscape, i.e. the collection of stable and metastable vacua obtained from flux compactifications on conformal Calabi-Yau manifolds. We show that monodromies are important for the topographic structure of the landscape. In particular we find that there are long series of continuously connected vacua in the complex structure moduli space of the internal manifold. We also use geometric transitions to connect the moduli spaces of different manifolds, and create longer series of vacua. Finally, we investigate the stability of string theory vacua by constructing semiclassical instantons. These results have implications for the population of the landscape by eternal inflation.

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