Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

Bayesian Inference for Retrospective Population Genetics Models Using Markov Chain Monte Carlo Methods

Genetics, the science of heredity and variation in living organisms, has a central role in medicine, in breeding crops and livestock, and in studying fundamental topics of biological sciences such as evolution and cell functioning. Currently the field of genetics is under a rapid development because of the recent advances in technologies by which molecular data can be obtained from living organisms. In order that most information from such data can be extracted, the analyses need to be carried out using statistical models that are tailored to take account of the particular genetic processes. In this thesis we formulate and analyze Bayesian models for genetic marker data of contemporary individuals. The major focus is on the modeling of the unobserved recent ancestry of the sampled individuals (say, for tens of generations or so), which is carried out by using explicit probabilistic reconstructions of the pedigree structures accompanied by the gene flows at the marker loci. For such a recent history, the recombination process is the major genetic force that shapes the genomes of the individuals, and it is included in the model by assuming that the recombination fractions between the adjacent markers are known. The posterior distribution of the unobserved history of the individuals is studied conditionally on the observed marker data by using a Markov chain Monte Carlo algorithm (MCMC). The example analyses consider estimation of the population structure, relatedness structure (both at the level of whole genomes as well as at each marker separately), and haplotype configurations. For situations where the pedigree structure is partially known, an algorithm to create an initial state for the MCMC algorithm is given. Furthermore, the thesis includes an extension of the model for the recent genetic history to situations where also a quantitative phenotype has been measured from the contemporary individuals. In that case the goal is to identify positions on the genome that affect the observed phenotypic values. This task is carried out within the Bayesian framework, where the number and the relative effects of the quantitative trait loci are treated as random variables whose posterior distribution is studied conditionally on the observed genetic and phenotypic data. In addition, the thesis contains an extension of a widely-used haplotyping method, the PHASE algorithm, to settings where genetic material from several individuals has been pooled together, and the allele frequencies of each pool are determined in a single genotyping.

Towards real-time understanding of processes in pharmaceutical powder technology

There is a need for better understanding of the processes and new ideas to develop traditional pharmaceutical powder manufacturing procedures. Process analytical technology (PAT) has been developed to improve understanding of the processes and establish methods to monitor and control processes. The interest is in maintaining and even improving the whole manufacturing process and the final products at real-time. Process understanding can be a foundation for innovation and continuous improvement in pharmaceutical development and manufacturing. New methods are craved for to increase the quality and safety of the final products faster and more efficiently than ever before. The real-time process monitoring demands tools, which enable fast and noninvasive measurements with sufficient accuracy. Traditional quality control methods have been laborious and time consuming and they are performed off line i.e. the analysis has been removed from process area. Vibrational spectroscopic methods are responding this challenge and their utilisation have increased a lot during the past few years. In addition, other methods such as colour analysis can be utilised in noninvasive real-time process monitoring. In this study three pharmaceutical processes were investigated: drying, mixing and tabletting. In addition tablet properties were evaluated. Real-time monitoring was performed with NIR and Raman spectroscopies, colour analysis, particle size analysis and compression data during tabletting was evaluated using mathematical modelling. These methods were suitable for real-time monitoring of pharmaceutical unit operations and increase the knowledge of the critical parameters in the processes and the phenomena occurring during operations. They can improve our process understanding and therefore, finally, enhance the quality of final products.

Plant secondary compounds and soil microbial processes in carbon and nitrogen cycling in relation to tree species

The aim of this study was to explore soil microbial activities related to C and N cycling and the occurrence and concentrations of two important groups of plant secondary compounds, terpenes and phenolic compounds, under silver birch (Betula pendula Roth), Norway spruce (Picea abies (L.) Karst) and Scots pine (Pinus sylvestris L.) as well as to study the effects of volatile monoterpenes and tannins on soil microbial activities. The study site, located in Kivalo, northern Finland, included ca. 70-year-old adjacent stands dominated by silver birch, Norway spruce and Scots pine. Originally the soil was very probably similar in all three stands. All forest floor layers (litter (L), fermentation layer (F) and humified layer (H)) under birch and spruce showed higher rates of CO2 production, greater net mineralisation of nitrogen and higher amounts of carbon and nitrogen in microbial biomass than did the forest floor layers under pine. Concentrations of mono-, sesqui-, di- and triterpenes were higher under both conifers than under birch, while the concentration of total water-soluble phenolic compounds as well as the concentration of condensed tannins tended to be higher or at least as high under spruce as under birch or pine. In general, differences between tree species in soil microbial activities and in concentrations of secondary compounds were smaller in the H layer than in the upper layers. The rate of CO2 production and the amount of carbon in the microbial biomass correlated highly positively with the concentration of total water-soluble phenolic compounds and positively with the concentration of condensed tannins. Exposure of soil to volatile monoterpenes and tannins extracted and fractionated from spruce and pine needles affected carbon and nitrogen transformations in soil, but the effects were dependent on the compound and its molecular structure. Monoterpenes decreased net mineralisation of nitrogen and probably had a toxic effect on part of the microbial population in soil, while another part of the microbes seemed to be able to use monoterpenes as a carbon source. With tannins, low-molecular-weight compounds (also compounds other than tannins) increased soil CO2 production and nitrogen immobilisation by soil microbes while the higher-molecular-weight condensed tannins had inhibitory effects. In conclusion, plant secondary compounds may have a great potential in regulation of C and N transformations in forest soils, but the real magnitude of their significance in soil processes is impossible to estimate.

Ecological processes and large-scale climate relationships in northern coniferous forests

Ilmasto vaikuttaa ekologisiin prosesseihin eri tasoilla. Suuren mittakaavan ilmastoprosessit, yhdessä ilmakehän ja valtamerien kanssa, säätelevät paikallisia sääilmiöitä suurilla alueilla (mantereista pallopuoliskoihin). Tämä väistöskirja pyrkii selittämään kuinka suuren mittakaavan ilmasto on vaikuttanut tiettyihin ekologisiin prosesseihin pohjoisella havumetsäalueella. Valitut prosessit olivat puiden vuosilustojen kasvu, metsäpalojen esiintyminen ja vuoristomäntykovakuoriaisen aiheuttamat puukuolemat. Suuren mittakaavan ilmaston löydettiin vaikuttaneen näiden prosessien esiintymistiheyteen, kestoon ja levinneisyyteen keskeisten sään muuttujien välityksellä hyvin laajoilla alueilla. Tutkituilla prosesseilla oli vahva yhteys laajan mittakaavan ilmastoon. Yhteys on kuitenkin ollut hyvin dynaaminen ja muuttunut 1900-luvulla ilmastonmuutoksen aiheuttaessa muutoksia suuren mittakaavan ja alueellisten ilmastoprosessien välisiin sisäisiin suhteisiin.

Integral Transformations of Volterra Gaussian Processes

We study integral representations of Gaussian processes with a pre-specified law in terms of other Gaussian processes. The dissertation consists of an introduction and of four research articles. In the introduction, we provide an overview about Volterra Gaussian processes in general, and fractional Brownian motion in particular. In the first article, we derive a finite interval integral transformation, which changes fractional Brownian motion with a given Hurst index into fractional Brownian motion with an other Hurst index. Based on this transformation, we construct a prelimit which formally converges to an analogous, infinite interval integral transformation. In the second article, we prove this convergence rigorously and show that the infinite interval transformation is a direct consequence of the finite interval transformation. In the third article, we consider general Volterra Gaussian processes. We derive measure-preserving transformations of these processes and their inherently related bridges. Also, as a related result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale. In the fourth article, we derive a class of ergodic transformations of self-similar Volterra Gaussian processes.

Modeling ecological and evolutionary processes in spatially structured populations

Many species inhabit fragmented landscapes, resulting either from anthropogenic or from natural processes. The ecological and evolutionary dynamics of spatially structured populations are affected by a complex interplay between endogenous and exogenous factors. The metapopulation approach, simplifying the landscape to a discrete set of patches of breeding habitat surrounded by unsuitable matrix, has become a widely applied paradigm for the study of species inhabiting highly fragmented landscapes. In this thesis, I focus on the construction of biologically realistic models and their parameterization with empirical data, with the general objective of understanding how the interactions between individuals and their spatially structured environment affect ecological and evolutionary processes in fragmented landscapes. I study two hierarchically structured model systems, which are the Glanville fritillary butterfly in the Åland Islands, and a system of two interacting aphid species in the Tvärminne archipelago, both being located in South-Western Finland. The interesting and challenging feature of both study systems is that the population dynamics occur over multiple spatial scales that are linked by various processes. My main emphasis is in the development of mathematical and statistical methodologies. For the Glanville fritillary case study, I first build a Bayesian framework for the estimation of death rates and capture probabilities from mark-recapture data, with the novelty of accounting for variation among individuals in capture probabilities and survival. I then characterize the dispersal phase of the butterflies by deriving a mathematical approximation of a diffusion-based movement model applied to a network of patches. I use the movement model as a building block to construct an individual-based evolutionary model for the Glanville fritillary butterfly metapopulation. I parameterize the evolutionary model using a pattern-oriented approach, and use it to study how the landscape structure affects the evolution of dispersal. For the aphid case study, I develop a Bayesian model of hierarchical multi-scale metapopulation dynamics, where the observed extinction and colonization rates are decomposed into intrinsic rates operating specifically at each spatial scale. In summary, I show how analytical approaches, hierarchical Bayesian methods and individual-based simulations can be used individually or in combination to tackle complex problems from many different viewpoints. In particular, hierarchical Bayesian methods provide a useful tool for decomposing ecological complexity into more tractable components.

Below-ground processes in meadow soil under elevated ozone and carbon dioxide : - Greenhouse gas fluxes, N cycling and microbial communities

This thesis focuses on how elevated CO2 and/or O3 affect the below-ground processes in semi-natural vegetation, with an emphasis on greenhouse gases, N cycling and microbial communities. Meadow mesocosms mimicking lowland hay meadows in Jokioinen, SW Finland, were enclosed in open-top chambers and exposed to ambient and elevated levels of O3 (40-50 ppb) and/or CO2 (+100 ppm) for three consecutive growing season, while chamberless plots were used as chamber controls. Chemical and microbiological analyses as well as laboratory incubations of the mesocosm soils under different treatments were used to study the effects of O3 and/or CO2. Artificially constructed mesocosms were also compared with natural meadows with regards to GHG fluxes and soil characteristics. In addition to research conducted at the ecosystem level (i.e. the mesocosm study), soil microbial communities were also examined in a pot experiment with monocultures of individual species. By comparing mesocosms with similar natural plant assemblage, it was possible to demonstrate that artificial mesocosms simulated natural habitats, even though some differences were found in the CH4 oxidation rate, soil mineral N, and total C and N concentrations in the soil. After three growing seasons of fumigations, the fluxes of N2O, CH4, and CO2 were decreased in the NF+O3 treatment, and the soil NH4+-N and mineral N concentrations were lower in the NF+O3 treatment than in the NF control treatment. The mesocosm soil microbial communities were affected negatively by the NF+O3 treatment, as the total, bacterial, actinobacterial, and fungal PLFA biomasses as well as the fungal:bacterial biomass ratio decreased under elevated O3. In the pot survey, O3 decreased the total, bacterial, actinobacterial, and mycorrhizal PLFA biomasses in the bulk soil and affected the microbial community structure in the rhizosphere of L. pratensis, whereas the bulk soil and rhizosphere of the other monoculture, A. capillaris, remained unaffected by O3. Elevated CO2 caused only minor and insignificant changes in the GHG fluxes, N cycling, and the microbial community structure. In the present study, the below-ground processes were modified after three years of moderate O3 enhancement. A tentative conclusion is that a decrease in N availability may have feedback effects on plant growth and competition and affect the N cycling of the whole meadow ecosystem. Ecosystem level changes occur slowly, and multiplication of the responses might be expected in the long run.

Behavioural, population, and genetic processes affecting metapopulation dynamics of the Glanville fritillary butterfly

In my thesis I have been studying the effects of population fragmentation and extinction-recolonization dynamics on genetic and evolutionary processes in the Glanville fritillary butterfly (Melitaea cinxia). By conducting crosses within and among newly-colonized populations and using several fitness measures, I found a strong decrease in fitness following colonization by a few related individuals, and a strong negative relationship between parental relatedness and offspring fitness. Thereafter, I was interested in determining the number and relatedness of individuals colonizing new populations, which I did using a set of microsatellites I had previously developed for this species. Additionally, I am interested in the evolution of key life-history traits. By following the lifetime reproductive success of males emerging at different times in a semi-natural setup, I demonstrated that protandry is adaptive in males, and I was able to rule out, for M. cinxia, alternative incidental hypotheses evoked to explain the evolution of protandry in insects. Finally, in work I did together with Prof. Hanna Kokko, I am proposing bet-hedging as a new mechanism that could explain the evolution of polyandry in M. cinxia.

Aerosol Particle Formation: Meteorological and Synoptic Processes behind the Event

Aerosol particles in the atmosphere are known to significantly influence ecosystems, to change air quality and to exert negative health effects. Atmospheric aerosols influence climate through cooling of the atmosphere and the underlying surface by scattering of sunlight, through warming of the atmosphere by absorbing sun light and thermal radiation emitted by the Earth surface and through their acting as cloud condensation nuclei. Aerosols are emitted from both natural and anthropogenic sources. Depending on their size, they can be transported over significant distances, while undergoing considerable changes in their composition and physical properties. Their lifetime in the atmosphere varies from a few hours to a week. New particle formation is a result of gas-to-particle conversion. Once formed, atmospheric aerosol particles may grow due to condensation or coagulation, or be removed by deposition processes. In this thesis we describe analyses of air masses, meteorological parameters and synoptic situations to reveal conditions favourable for new particle formation in the atmosphere. We studied the concentration of ultrafine particles in different types of air masses, and the role of atmospheric fronts and cloudiness in the formation of atmospheric aerosol particles. The dominant role of Arctic and Polar air masses causing new particle formation was clearly observed at Hyytiälä, Southern Finland, during all seasons, as well as at other measurement stations in Scandinavia. In all seasons and on multi-year average, Arctic and North Atlantic areas were the sources of nucleation mode particles. In contrast, concentrations of accumulation mode particles and condensation sink values in Hyytiälä were highest in continental air masses, arriving at Hyytiälä from Eastern Europe and Central Russia. The most favourable situation for new particle formation during all seasons was cold air advection after cold-front passages. Such a period could last a few days until the next front reached Hyytiälä. The frequency of aerosol particle formation relates to the frequency of low-cloud-amount days in Hyytiälä. Cloudiness of less than 5 octas is one of the factors favouring new particle formation. Cloudiness above 4 octas appears to be an important factor that prevents particle growth, due to the decrease of solar radiation, which is one of the important meteorological parameters in atmospheric particle formation and growth. Keywords: Atmospheric aerosols, particle formation, air mass, atmospheric front, cloudiness