On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution

A birth-death process is a continuous-time Markov chain that counts the number of particles in a system over time. In the general process with $n$ current particles, a new particle is born with instantaneous rate $\lambda_n$ and a particle dies with instantaneous rate $\mu_n$. Currently no robust and efficient method exists to evaluate the finite-time transition probabilities in a general birth-death process with arbitrary birth and death rates. In this paper, we first revisit the theory of continued fractions to obtain expressions for the Laplace transforms of these transition probabilities and make explicit an important derivation connecting transition probabilities and continued fractions. We then develop an efficient algorithm for computing these probabilities that analyzes the error associated with approximations in the method. We demonstrate that this error-controlled method agrees with known solutions and outperforms previous approaches to computing these probabilities. Finally, we apply our novel method to several important problems in ecology, evolution, and genetics

Frailty effects in networks: comparison and identification of individual heterogeneity versus preferential attachment in evolving networks

Preferential attachment is a proportionate growth process in networks, where nodes receive new links in proportion to their current degree. Preferential attachment is a popular generative mechanism to explain the widespread observation of power-law-distributed networks. An alternative explanation for the phenomenon is a randomly grown network with large individual variation in growth rates among the nodes (frailty). We derive analytically the distribution of individual rates, which will reproduce the connectivity distribution that is obtained from a general preferential attachment process (Yule process), and the structural differences between the two types of graphs are examined by simulations. We present a statistical test to distinguish the two generative mechanisms from each other and we apply the test to both simulated data and two real data sets of scientific citation and sexual partner networks. The findings from the latter analyses argue for frailty effects as an important mechanism underlying the dynamics of complex networks

Tarjouskirjan stokastinen mallintaminen

Tässä kandidaatintyössä käsitellään tarjouskirjan stokastista mallintamista, mikä yhdistää useita eri matematiikan osa-alueita. Työssä käsitellään yhtä keskeistä mallia syvällisesti, mikä johtaa ymmärrykseen tarjouskirjan mallintamisesta ja mallin kritisointiin empiiristen tutkimusten ja vaihtoehtoisten mallien pohjalta. Työ on toteutettu kirjallisuuskatsauksena eli malleja on vertailtu kvalitatiivisesti ja selkeitä kehityskohteita on esitetty muiden tutkimusten pohjalta. Tarjouskirjan mallintamisessa tärkeitä mallinnettavia asioita ovat osto- ja myyntihinta, tarjouksien määrät eri hinnoilla ja todennäköisyydet, joilla muutoksia tapahtuu tarjouskirjassa. Työssä tarjouskirja on ensin esitetty matemaattisin merkinnöin, minkä jälkeen näille tärkeille ominaisuuksille on esitetty matemaattiset kaavat tulevaisuuteen. Samalla malliin on esitetty kehityskohteita, kuten esimerkiksi parametrien estimoinnissa vaihtoehtoisia tapoja löytää parhaat parametrit. Työssä esitetään matemaattiset laskukaavat todennäköisyyksille, joilla keskihinta nousee tai laskee, rajahintatarjous toteutuu ennen hinnan muuttumista ja markkinantakaaja tuottaa hintaeron. Näiden todennäköisyyksien esittäminen mahdollistaa eri kaupankäyntistrategioiden kehittämisen ja työssä esitellään näistä kaksi yksinkertaista vaihtoehtoa. Ensin työssä esitetään miten osake kannattaa ostaa ja myydä myöhemmin, jos hinnan nousemiselle on korkea todennäköisyys, ja tämän jälkeen hintaeron tuottamiselle esitetään strategia. Mallin vertaaminen vaihtoehtoisiin malleihin on toteutettu etsimällä mahdollisimman monipuolisia tapoja mallintaa tarjouskirjaa. Tässä ideana on ollut löytää eri kehityskohteita esitettyyn malliin, jotta mahdollinen suurin syy epätarkalle tulokselle voitaisiin löytää. Vaihtoehtoisista malleista löydettiin eroavaisuuksia tarjouksien yksittäisestä tarkastelusta, tapahtumien välisistä korrelaatioista, differentiaaliyhtälöiden tärkeydestä, oletetusta jakaumasta, ratkaisun analyyttisyydestä, volatiliteetin vaikutuksesta, taloustieteellisestä selityksestä ja omien toimintojen vaikutuksesta tarjouskirjaan. Näiden havaintojen perusteella malliin mietittiin mahdollisia kehityskohteita, jotka olisivat jatkotutkimuksen kohteina.This bachelor’s thesis studies stochastic order book modelling, which includes many different subjects from mathematics. The thesis studies a single stochastic order book model more deeply, which helps with understanding order book modelling and in criticizing the model. The criticizing is done by comparing the model to other models and empirical studies. The bachelor’s thesis is a literature review, and thus it is qualitative in nature and the ideas for further improvements are based on other studies. The most important values in order book modelling are ask and bid prices, number of orders on each price, and probabilities of different changes in the order book. In the thesis, the order book is first introduced with mathematical notation, which is followed by the equations of different probabilities of changes. Furthermore, while the order book is introduced, different improvement ideas are introduced. For example, when estimating parameters for the model, one could use different methods to get better results. The three equations of probabilities are introduced for increase in mid-price, executing order before mid-price moves and making the spread. The introduction of the equations makes it possible to use them in simple trading strategies of which two are introduced. In the first one, a market participant buys a stock and sells it later if there is a high probability of increase in the mid-price. In the second one, a market participant should enter two limit book orders if there is a high probability of making the spread. Different models are briefly introduced to compare them to the main model. The different models are as different as possible to get maximum utility from them for the main model. The different models differ in the use of unit size in order sizes, the correlations between different changes, the importance of differential equations, the assumption of the probability distribution, the analytical solution, the effect of volatility, the economic explanation and how a market participant’s actions affect the order book. These observations were used to make propositions for further improvements in the model

First passage events in biological systems with non-exponential inter-event times

It is often possible to model the dynamics of biological systems as a series of discrete transitions between a finite set of observable states (or compartments). When the residence times in each state, or inter-event times more generally, are exponentially distributed, then one can write a set of ordinary differential equations, which accurately describe the evolution of mean quantities. Non-exponential inter-event times can also be experimentally observed, but are more difficult to analyse mathematically. In this paper, we focus on the computation of first passage events and their probabilities in biological systems with non-exponential inter-event times. We show, with three case studies from Molecular Immunology, Virology and Epidemiology, that significant errors are introduced when drawing conclusions based on the assumption that inter-event times are exponentially distributed. Our approach allows these errors to be avoided with the use of phase-type distributions that approximate arbitrarily distributed inter-event times

Rational approximation schemes for solutions of abstract Cauchy problems and evolution equations

In this dissertation we study time and space discretization methods for approximating solutions of abstract Cauchy problems and evolution equations in a Banach space setting. Two extensions of the Hille-Phillips functional calculus are developed. The first result is the Hille-Phillips functional calculus for generators of bi-continuous semigroups, and the second is a C-regularized version of the Hille-Phillips functional calculus for generators of C-regularized semigroups. These results are used in order to study time discretization schemes for abstract Cauchy problems associated with generators of bi-continuous semigroups as well as C-regularized semigoups. Stability, convergence results, and error estimates for rational approximation schemes for bi-continuous and C-regularized semigroups are presented. We also extend the Trotter-Kato Theorem to the framework of C-regularized semigroups and combine it with the time discretization methods previously mentioned in order to obtain fully discretized schemes, provided by A-stable rational functions. Among the applications, we outline how to use rational approximation schemes to approximate solutions of nonlinear ODE\u27s, and we show the significance of the results for bi-continuous semigroups for obtaining new numerical inversion formulas for the Laplace transform (with sharp error estimates). Furthermore, rational approximation schemes for integrated semigroups are presented with applications to the second order abstract Cauchy problem