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.

Aspects of Inflationary Models at Low Energy Scales

Cosmological inflation is the dominant paradigm in explaining the origin of structure in the universe. According to the inflationary scenario, there has been a period of nearly exponential expansion in the very early universe, long before the nucleosynthesis. Inflation is commonly considered as a consequence of some scalar field or fields whose energy density starts to dominate the universe. The inflationary expansion converts the quantum fluctuations of the fields into classical perturbations on superhorizon scales and these primordial perturbations are the seeds of the structure in the universe. Moreover, inflation also naturally explains the high degree of homogeneity and spatial flatness of the early universe. The real challenge of the inflationary cosmology lies in trying to establish a connection between the fields driving inflation and theories of particle physics. In this thesis we concentrate on inflationary models at scales well below the Planck scale. The low scale allows us to seek for candidates for the inflationary matter within extensions of the Standard Model but typically also implies fine-tuning problems. We discuss a low scale model where inflation is driven by a flat direction of the Minimally Supersymmetric Standard Model. The relation between the potential along the flat direction and the underlying supergravity model is studied. The low inflationary scale requires an extremely flat potential but we find that in this particular model the associated fine-tuning problems can be solved in a rather natural fashion in a class of supergravity models. For this class of models, the flatness is a consequence of the structure of the supergravity model and is insensitive to the vacuum expectation values of the fields that break supersymmetry. Another low scale model considered in the thesis is the curvaton scenario where the primordial perturbations originate from quantum fluctuations of a curvaton field, which is different from the fields driving inflation. The curvaton gives a negligible contribution to the total energy density during inflation but its perturbations become significant in the post-inflationary epoch. The separation between the fields driving inflation and the fields giving rise to primordial perturbations opens up new possibilities to lower the inflationary scale without introducing fine-tuning problems. The curvaton model typically gives rise to relatively large level of non-gaussian features in the statistics of primordial perturbations. We find that the level of non-gaussian effects is heavily dependent on the form of the curvaton potential. Future observations that provide more accurate information of the non-gaussian statistics can therefore place constraining bounds on the curvaton interactions.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Applications of dimensional reduction to electroweak and QCD matter

When heated to high temperatures, the behavior of matter changes dramatically. The standard model fields go through phase transitions, where the strongly interacting quarks and gluons are liberated from their confinement to hadrons, and the Higgs field condensate melts, restoring the electroweak symmetry. The theoretical framework for describing matter at these extreme conditions is thermal field theory, combining relativistic field theory and quantum statistical mechanics. For static observables the physics is simplified at very high temperatures, and an effective three-dimensional theory can be used instead of the full four-dimensional one via a method called dimensional reduction. In this thesis dimensional reduction is applied to two distinct problems, the pressure of electroweak theory and the screening masses of mesonic operators in quantum chromodynamics (QCD). The introductory part contains a brief review of finite-temperature field theory, dimensional reduction and the central results, while the details of the computations are contained in the original research papers. The electroweak pressure is shown to converge well to a value slightly below the ideal gas result, whereas the pressure of the full standard model is dominated by the QCD pressure with worse convergence properties. For the mesonic screening masses a small positive perturbative correction is found, and the interpretation of dimensional reduction on the fermionic sector is discussed.

Rhetoric and Representation : Exploring the Cultural Meaning of the Natural Sciences in Contemporary Popular Science Writing and Literature

Kirjallisuuden- ja kulttuurintutkimus on viimeisten kolmen vuosikymmenen aikana tullut yhä enenevässä määrin tietoiseksi tieteen ja taiteen suhteen monimutkaisesta luonteesta. Nykyään näiden kahden kulttuurin tutkimus muodostaa oman kenttänsä, jolla niiden suhdetta tarkastellaan ennen kaikkea dynaamisena vuorovaikutuksena, joka heijastaa kulttuurimme kieltä, arvoja ja ideologisia sisältöjä. Toisin kuin aiemmat näkemykset, jotka pitävät tiedettä ja taidetta toisilleen enemmän tai vähemmän vastakkaisina pyrkimyksinä, nykytutkimus lähtee oletuksesta, jonka mukaan ne ovat kulttuurillisesti rakentuneita diskursseja, jotka kohtaavat usein samankaltaisia todellisuuden mallintamiseen liittyviä ongelmia, vaikka niiden käyttämät metodit eroavatkin toisistaan. Väitöskirjani keskittyy yllä mainitun suhteen osa-alueista popularisoidun tietokirjallisuuden (muun muassa Paul Davies, James Gleick ja Richard Dawkins) käyttämän kielen ja luonnontieteistä ideoita ammentavan kaunokirjallisuuden (muun muassa Jeanette Winterson, Tom Stoppard ja Richard Powers) hyödyntämien keinojen tarkasteluun nojautuen yli 30 teoksen kattavaa aineistoa koskevaan tyylin ja teemojen tekstianalyysiin. Populaarin tietokirjallisuuden osalta tarkoituksenani on osoittaa, että sen käyttämä kieli rakentuu huomattavassa määrin sellaisille rakenteille, jotka tarjoavat mahdollisuuden esittää todellisuutta koskevia argumentteja mahdollisimman vakuuttavalla tavalla. Tässä tehtävässä monilla klassisen retoriikan määrittelemillä kuvioilla on tärkeä rooli, koska ne auttavat liittämään sanotun sisällön ja muodon tiukasti toisiinsa: retoristen kuvioiden käyttö ei näin ollen edusta pelkkää tyylikeinoa, vaan se myös usein kiteyttää argumenttien taustalla olevat tieteenfilosofiset olettamukset ja auttaa vakiinnuttamaan argumentoinnin logiikan. Koska monet aikaisemmin ilmestyneistä tutkimuksista ovat keskittyneet pelkästään metaforan rooliin tieteellisissä argumenteissa, tämä väitöskirja pyrkii laajentamaan tutkimuskenttää analysoimalla myös toisenlaisten kuvioiden käyttöä. Osoitan myös, että retoristen kuvioiden käyttö muodostaa yhtymäkohdan tieteellisiä ideoita hyödyntävään kaunokirjallisuuteen. Siinä missä popularisoitu tiede käyttää retoriikkaa vahvistaakseen sekä argumentatiivisia että kaunokirjallisia ominaisuuksiaan, kuvaa tällainen sanataide tiedettä tavoilla, jotka usein heijastelevat tietokirjallisuuden kielellisiä rakenteita. Toisaalta on myös mahdollista nähdä, miten kaunokirjallisuuden keinot heijastuvat popularisoidun tieteen kerrontatapoihin ja kieleen todistaen kahden kulttuurin dynaamisesta vuorovaikutuksesta. Nykyaikaisen populaaritieteen retoristen elementtien ja kaunokirjallisuuden keinojen vertailu näyttää lisäksi, kuinka tiede ja taide osallistuvat keskusteluun kulttuurimme tiettyjen peruskäsitteiden kuten identiteetin, tiedon ja ajan merkityksestä. Tällä tavoin on mahdollista nähdä, että molemmat ovat perustavanlaatuisia osia merkityksenantoprosessissa, jonka kautta niin tieteelliset ideat kuin ihmiselämän suuret kysymyksetkin saavat kulttuurillisesti rakentuneen merkityksensä.

Patterns of Transcendence : Classical Myth in Marina Tsvetaeva's Poetry of the 1920s

This study examines the position and meaning of Classical mythological plots, themes and characters in the oeuvre of the Russian Modernist poet Marina Tsvetaeva (1892-1941). The material consists of lyric poems from the collection Posle Rossii (1928) and two longer lyrical tragedies, Ariadna (1924) and Fedra (1927). These works are examined in the context of Russian Modernism and Tsvetaeva s own poetic development, also taking into account the author s biography, namely, her correspondence with Boris Pasternak. Tsvetaeva s appropriations of the myths enter into a dialogue with the Classical tradition and with the earlier Russian and Western literary manifestations of the source material. Her Classical texts are inextricably linked with her own authorial myth, they are used to project both her ideas about poetry as well as the authored self of her poems. An important context for Tsvetaeva s application of the Classical myths is the concept of the Platonic ladder of Eros. This plot evokes the process of transcendence of the mortal subject into the immaterial realm and is applied by the author as an extended metaphor of the poet s birth. Emphasizing the dialectical movement between the earthly and the divine, Tsvetaeva s Classical personae foreground various positions of the individual between these two realms. By means of kaleidoscopic reformulations of similar metaphors and concepts, Tsvetaeva s mythological poems illustrate the poet s position between the material and the immaterial and the various consequences of this dichotomy on the creative mission. At the heart of Tsvetaeva s appropriation of the Sibyl, Phaedra, Eurydice and Ariadne is the tension between the body and disembodiment. The two lyrical tragedies develop the dichotomous worldview further, nevertheless emphasizing the dual perspective of the divine and the earthly realms: immaterial existence is often evaluated from a material perspective and vice versa. The Platonic subtext is central for Ariadna, focussing on Theseus development from an earthly hero to a spiritual one. Fedra concentrates on Phaedra s divinely induced physical passion, which is nevertheless evoked in a creative light.