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This thesis focuses on measuring extreme risks in insurance business. We mainly use extreme value theory to develop asymptotics for risk measures. We also study the characterization of upper comonotonicity for multiple extreme risks. Firstly, we conduct asymptotics for the Haezendonck--Goovaerts (HG) risk measure of extreme risks at high confidence levels, which serves as an alternative way to statistical simulations. We split the study of this problem into two steps. In the first step, we concentrate on the HG risk measure with a power Young function, which yields certain explicitness. Then we derive asymptotics for a risk variable with a distribution function that belongs to one of the three max-domains of attraction separately. We extend our asymptotic study to the HG risk measure with a general Young function in the second step. We study this problem using different approaches and overcome a lot of technical difficulties. The risk variable is assumed to follow a distribution function that belongs to the max-domain of attraction of the generalized extreme value distribution and we show a unified proof for all three max-domains of attraction. Secondly, we study the first- and second-order asymptotics for the tail distortion risk measure of extreme risks. Similarly as in the first part, we develop the first-order asymptotics for the tail distortion risk measure of a risk variable that follows a distribution function belonging to the max-domain of attraction of the generalized extreme value distribution. In order to improve the accuracy of the first-order asymptotics, we further develop the second-order asymptotics for the tail distortion risk measure. Numerical examples are carried out to show the accuracy of both asymptotics and the great improvements of the second-order asymptotics. Lastly, we characterize the upper comonotonicity via tail convex order. For any given marginal distributions, a maximal random vector with respect to tail convex order is proved to be upper comonotonic under suitable conditions. As an application, we consider the computation of the HG risk measure of the sum of upper comonotonic random variables with exponential marginal distributions. The methodology developed in this thesis is expected to work with the same efficiency for generalized quantiles (such as expectile, Lp-quantiles, ML-quantiles and Orlicz quantiles), quantile based risk measures or risk measures which focus on the tail areas, and also work well on capital allocation problems.

This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

This thesis investigates implied volatility in general classes of stock price models. To begin with, we take a very general view. We find that implied volatility is always, everywhere, and for every expiry well-defined only if the stock price is a non-negative martingale. We also derive sufficient and close to necessary conditions for an implied volatility surface to be free from static arbitrage. In this context, free from static arbitrage means that the call price surface generated by the implied volatility surface is free from static arbitrage. We also investigate the small time to expiry behaviour of implied volatility. We do this in almost complete generality, assuming only that the call price surface is non-decreasing and right continuous in time to expiry and that the call surface satisfies the no-arbitrage bounds (S-K)+≤ C(K, τ)≤ S. We used S to denote the current stock price, K to be a option strike price, τ denotes time to expiry, and C(K, τ) the price of the K strike option expiring in τ time units. Under these weak assumptions, we obtain exact asymptotic formulae relating the call price surface and the implied volatility surface close to expiry. We apply our general asymptotic formulae to determining the small time to expiry behaviour of implied volatility in a variety of models. We consider exponential L??vy models, obtaining new and somewhat surprising results. We then investigate the behaviour close to expiry of stochastic volatility models in the at-the-money case. Our results generalise what is already known and by a novel method of proof. In the not at-the-money case, we consider local volatility models using classical results of Varadhan. In obtaining the asymptotics for local volatility models, we use a representation of the European call as an integral over time to expiry. We devote an entire chapter to representations of the European call option; a key role is played by local time and the argument of Klebaner. A novel alternative that is especially useful in the local volatility case is also presented.

abstract: This work is concerned with the study and numerical solution of large reaction diffusion systems with applications to the simulation of degradation effects in solar cells. A discussion of the basics of solar cells including the function of solar cells, the degradation of energy efficiency that happens over time, defects that affect solar cell efficiency and specific defects that can be modeled with a reaction diffusion system are included. Also included is a simple model equation of a solar cell. The basics of stoichiometry theory, how it applies to kinetic reaction systems, and some conservation properties are introduced. A model that considers the migration of defects in addition to the reaction processes is considered. A discussion of asymptotics and how it relates to the numerical simulation of the lifetime of solar cells is included. A reduced solution is considered and a presentation of a numerical comparison of the reduced solution with the full solution on a simple test problem is included. Operator splitting techniques are introduced and discussed. Asymptotically preserving schemes combine asymptotics and operator splitting to use reasonable time steps. A presentation of a realistic example of this study applied to solar cells follows. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

Shock waves in reactive media possess very rich dynamics: from formation of cells in multiple dimensions to oscillating shock fronts in one-dimension. Because of the extreme complexity of the equations of combustion theory, most of the current understanding of unstable detonation waves relies on extensive numerical simulations of the reactive compressible Euler/Navier-Stokes equations. Attempts at a simplified theory have been made in the past, most of which are very successful in describing steady detonation waves. In this work we focus on obtaining simplified theories capable of capturing not only the steady, but also the unsteady behavior of detonation waves. The first part of this thesis is focused on qualitative theories of detonation, where ad hoc models are proposed and analyzed. We show that equations as simple as a forced Burgers equation can capture most of the complex phenomena observed in detonations. In the second part of this thesis we focus on rational theories, and derive a weakly nonlinear model of multi-dimensional detonations. We also show, by analysis and numerical simulations, that the asymptotic equations provide good quantitative predictions.

No description available.

abstract: Fundamental limits of fixed-to-variable (F-V) and variable-to-fixed (V-F) length universal source coding at short blocklengths is characterized. For F-V length coding, the Type Size (TS) code has previously been shown to be optimal up to the third-order rate for universal compression of all memoryless sources over finite alphabets. The TS code assigns sequences ordered based on their type class sizes to binary strings ordered lexicographically. Universal F-V coding problem for the class of first-order stationary, irreducible and aperiodic Markov sources is first considered. Third-order coding rate of the TS code for the Markov class is derived. A converse on the third-order coding rate for the general class of F-V codes is presented which shows the optimality of the TS code for such Markov sources. This type class approach is then generalized for compression of the parametric sources. A natural scheme is to define two sequences to be in the same type class if and only if they are equiprobable under any model in the parametric class. This natural approach, however, is shown to be suboptimal. A variation of the Type Size code is introduced, where type classes are defined based on neighborhoods of minimal sufficient statistics. Asymptotics of the overflow rate of this variation is derived and a converse result establishes its optimality up to the third-order term. These results are derived for parametric families of i.i.d. sources as well as Markov sources. Finally, universal V-F length coding of the class of parametric sources is considered in the short blocklengths regime. The proposed dictionary which is used to parse the source output stream, consists of sequences in the boundaries of transition from low to high quantized type complexity, hence the name Type Complexity (TC) code. For large enough dictionary, the $\epsilon$-coding rate of the TC code is derived and a converse result is derived showing its optimality up to the third-order term. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2018

This thesis is concerned with extensions and refinements of the Katznelson-Tzafriri theorem, a cornerstone of the asymptotic theory of operator semigroups which recently has received renewed interest in the context of damped wave equations. The thesis comprises three main parts. The key results in the first part are a version of the Katznelson-Tzafriri theorem for bounded C_0-semigroups in which a certain function appearing in the original statement of the result is allowed more generally to be a bounded Borel measure, and bounds on the rate of decay in an important special case. The second part deals with the discrete version of the Katznelson-Tzafriri theorem and establishes upper and lower bounds on the rate of decay in this setting too. In an important special case these general bounds are then shown to be optimal for general Banach spaces but not on Hilbert space. The third main part, finally, turns to general operator semigroups. It contains a version of the Katznelson-Tzafriri theorem in the Hilbert space setting which relaxes the main assumption of the original result. Various applications and extensions of this general result are also presented.