Application of theoretical models to financial innovation

The present paper is aimed at providing a general strategic overview of the existing theoretical models that have applications in the field of financial innovation. Whereas most financialdevelopments have relied upon traditional economic tools, a new stream of research is defining a novel paradigm in which mathematical models from diverse scientific disciplines are being applied to conceptualize and explain economic and financial behavior. Indeed, terms such as ‘econophysics’ or ‘quantum finance’ have recently appeared to embrace efforts in this direction. As a first contact with such research, the project will present a brief description of some of the main theoretical models that have applications in finance and economics, and will try to present, if possible, potential new applications to particular areas in financial analysis, or new applicable models. As a result, emphasiswill be put on the implications of this research for the financial sector and its future dynamics.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Asymmetric stochastic switching driven by intrinsic molecular noise

Low-copy-number molecules are involved in many functions in cells. The intrinsic fluctuations of these numbers can enable stochastic switching between multiple steady states, inducing phenotypic variability. Herein we present a theoretical and computational study based on Master Equations and Fokker-Planck and Langevin descriptions of stochastic switching for a genetic circuit of autoactivation. We show that in this circuit the intrinsic fluctuations arising from low-copy numbers, which are inherently state-dependent, drive asymmetric switching. These theoretical results are consistent with experimental data that have been reported for the bistable system of the gallactose signaling network in yeast. Our study unravels that intrinsic fluctuations, while not required to describe bistability, are fundamental to understand stochastic switching and the dynamical relative stability of multiple states.

The IST METRA Project

This article summarizes the main achievementsof the Multi-Element Transmit andReceive Antennas (METRA) Project, an ISTresearch and technological development project carried out between January 2000 and June 2001 by Universitat Politècnica de Catalunya, the Center for Personkommunikation of Aalborg University, Nokia Networks, Nokia Mobile Phones, and Vodafone Group Research and Development.The main objective of METRA was the performanceevaluation of multi-antenna terminals incombination with adaptive antennas at the basestation in UMTS communication systems. 1 AMIMO channel sounder was developed that providedrealistic multi-antenna channel measurements.Using these measured data, stochasticchannel models were developed and properly validated.These models were also evaluated inorder to estimate their corresponding channelcapacity. Different MIMO configurations andprocessing schemes were developed for both theFDD and TDD modes of UTRA, and their linkperformance was assessed. Performance evaluationwas completed by system simulations thatillustrated the benefits of MIMO configurationsto the network operator. Implementation cost vs.performance improvement was also covered bythe project, including the base station and terminalmanufacturer and network operator viewpoints.Finally, significant standards contributionswere generated by the project and presented to the pertinent 3GPP working groups.

Frames: a maximum entropy statistical estimate of the inverse problem

A maximum entropy statistical treatment of an inverse problem concerning frame theory is presented. The problem arises from the fact that a frame is an overcomplete set of vectors that defines a mapping with no unique inverse. Although any vector in the concomitant space can be expressed as a linear combination of frame elements, the coefficients of the expansion are not unique. Frame theory guarantees the existence of a set of coefficients which is “optimal” in a minimum norm sense. We show here that these coefficients are also “optimal” from a maximum entropy viewpoint.

Effectiveness of early detection on breast cancer mortality reduction in Catalonia (Spain)

Background: At present, it is complicated to use screening trials to determine the optimal age intervals and periodicities of breast cancer early detection. Mathematical models are an alternative that has been widely used. The aim of this study was to estimate the effect of different breast cancer early detection strategies in Catalonia (Spain), in terms of breast cancer mortality reduction (MR) and years of life gained (YLG), using the stochastic models developed by Lee and Zelen (LZ). Methods: We used the LZ model to estimate the cumulative probability of death for a cohort exposed to different screening strategies after T years of follow-up. We also obtained the cumulative probability of death for a cohort with no screening. These probabilities were used to estimate the possible breast cancer MR and YLG by age, period and cohort of birth. The inputs of the model were: incidence of, mortality from and survival after breast cancer, mortality from other causes, distribution of breast cancer stages at diagnosis and sensitivity of mammography. The outputs were relative breast cancer MR and YLG. Results: Relative breast cancer MR varied from 20% for biennial exams in the 50 to 69 age interval to 30% for annual exams in the 40 to 74 age interval. When strategies differ in periodicity but not in the age interval of exams, biennial screening achieved almost 80% of the annual screening MR. In contrast to MR, the effect on YLG of extending screening from 69 to 74 years of age was smaller than the effect of extending the screening from 50 to 45 or 40 years. Conclusion: In this study we have obtained a measure of the effect of breast cancer screening in terms of mortality and years of life gained. The Lee and Zelen mathematical models have been very useful for assessing the impact of different modalities of early detection on MR and YLG in Catalonia (Spain).

Two component systems: physiological effect of a third component

Signal transduction systems mediate the response and adaptation of organisms to environmental changes. In prokaryotes, this signal transduction is often done through Two Component Systems (TCS). These TCS are phosphotransfer protein cascades, and in their prototypical form they are composed by a kinase that senses the environmental signals (SK) and by a response regulator (RR) that regulates the cellular response. This basic motif can be modified by the addition of a third protein that interacts either with the SK or the RR in a way that could change the dynamic response of the TCS module. In this work we aim at understanding the effect of such an additional protein (which we call ‘‘third component’’) on the functional properties of a prototypical TCS. To do so we build mathematical models of TCS with alternative designs for their interaction with that third component. These mathematical models are analyzed in order to identify the differences in dynamic behavior inherent to each design, with respect to functionally relevant properties such as sensitivity to changes in either the parameter values or the molecular concentrations, temporal responsiveness, possibility of multiple steady states, or stochastic fluctuations in the system. The differences are then correlated to the physiological requirements that impinge on the functioning of the TCS. This analysis sheds light on both, the dynamic behavior of synthetically designed TCS, and the conditions under which natural selection might favor each of the designs. We find that a third component that modulates SK activity increases the parameter space where a bistable response of the TCS module to signals is possible, if SK is monofunctional, but decreases it when the SK is bifunctional. The presence of a third component that modulates RR activity decreases the parameter space where a bistable response of the TCS module to signals is possible.

Contribution of early detection and adjuvant treatments to breast cancer mortality reduction in Catalonia, Spain

Background: Reductions in breast cancer (BC) mortality in Western countries have been attributed to the use of screening mammography and adjuvant treatments. The goal of this work was to analyze the contributions of both interventions to the decrease in BC mortality between 1975 and 2008 in Catalonia. Methodology/Principal Findings: A stochastic model was used to quantify the contribution of each intervention. Age standardized BC mortality rates for calendar years 1975-2008 were estimated in four hypothetical scenarios: 1) Only screening, 2) Only adjuvant treatment, 3) Both interventions, and 4) No intervention. For the 30-69 age group, observed Catalan BC mortality rates per 100,000 women-year rose from 29.4 in 1975 to 38.3 in 1993, and afterwards continuously decreased to 23.2 in 2008. If neither of the two interventions had been used, in 2008 the estimated BC mortality would have been 43.5, which, compared to the observed BC mortality rate, indicates a 46.7% reduction. In 2008 the reduction attributable to screening was 20.4%, to adjuvant treatments was 15.8% and to both interventions 34.1%. Conclusions/Significance: Screening and adjuvant treatments similarly contributed to reducing BC mortality in Catalonia. Mathematical models have been useful to assess the impact of interventions addressed to reduce BC mortality that occurred over nearly the same periods.

Renewal equations for option pricing

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these equations for typical contract specifications, in a particular but exemplifying case. We also show how a formal general solution can be found for more exotic derivatives, and we compare prices for alternative models of the underlying. Finally, we recover the celebrated results for the Wiener process under certain limits.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Exit times in non-Markovian drifting continuous-time random-walk processes

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

Asymmetric stochastic switching driven by intrinsic molecular noise

Low-copy-number molecules are involved in many functions in cells. The intrinsic fluctuations of these numbers can enable stochastic switching between multiple steady states, inducing phenotypic variability. Herein we present a theoretical and computational study based on Master Equations and Fokker-Planck and Langevin descriptions of stochastic switching for a genetic circuit of autoactivation. We show that in this circuit the intrinsic fluctuations arising from low-copy numbers, which are inherently state-dependent, drive asymmetric switching. These theoretical results are consistent with experimental data that have been reported for the bistable system of the gallactose signaling network in yeast. Our study unravels that intrinsic fluctuations, while not required to describe bistability, are fundamental to understand stochastic switching and the dynamical relative stability of multiple states.

Monotonic continuous-time random walks with drift and stochastic reset events

In this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes. We focus our attention on monotonic continuous-time random walks with a constant drift: The process increases between the reset events, either by the effect of the random jumps, or by the action of the deterministic drift. As a result of all these combined factors interesting properties emerge, like the existence (for any drift strength) of a stationary transition probability density function, or the faculty of the model to reproduce power-law-like behavior. General formulas for two extreme statistics, the survival probability, and the mean exit time, are also derived. To corroborate in an independent way the results of the paper, Monte Carlo methods were used. These numerical estimations are in full agreement with the analytical predictions.

Language extinction and linguistic fronts

Language diversity has become greatly endangered in the past centuries owing to processes of language shift from indigenous languages to other languages that are seen as socially and economically more advantageous, resulting in the death or doom of minority languages. In this paper, we define a new language competition model that can describe the historical decline of minority languages in competition with more advantageous languages. We then implement this non-spatial model as an interaction term in a reactiondiffusion system to model the evolution of the two competing languages. We use the results to estimate the speed at which the more advantageous language spreads geographically, resulting in the shrinkage of the area of dominance of the minority language. We compare the results from our model with the observed retreat in the area of influence of the Welsh language in the UK, obtaining a good agreement between the model and the observed data

A measure of stability as a criterion for the verification and analysis of simulation models

The aim of this study is to define a new statistic, PVL, based on the relative distance between the likelihood associated with the simulation replications and the likelihood of the conceptual model. Our results coming from several simulation experiments of a clinical trial show that the PVL statistic range can be a good measure of stability to establish when a computational model verifies the underlying conceptual model. PVL improves also the analysis of simulation replications because only one statistic is associated with all the simulation replications. As well it presents several verification scenarios, obtained by altering the simulation model, that show the usefulness of PVL. Further simulation experiments suggest that a 0 to 20 % range may define adequate limits for the verification problem, if considered from the viewpoint of an equivalence test.