In [Precise Asymptotics for Robust Stochastic Volatility Models; Ann. Appl. Probab. 2021] we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to short-time and small noise formulae for option prices, using the framework [Bayer et al; A regularity structure for rough volatility; Math. Fin. 2020]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence.
An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in $\ln S$ and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant (Barlow et. al, 2017, Q. Jl Mech. Appl. Math, 70 (1), 21-48) in the form of a modified symmetric Pad\'e approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.
The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al.. The Problem (A) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive penalization terms $\mathcal{D}_{U}$, which may not have a divergence formulation. In Problem (B) the objective functional, associated via Fenchel conjugacy to the terms $\mathcal{D}_{U}$, is not any more linear, as in OT or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a nonlinear subhedging value. Our theory allows us to establish a nonlinear robust pricing-hedging duality, which covers a wide range of known robust results. We also focus on Wasserstein-induced penalizations and we study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.
An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et. al, 2017, Q. Jl Mech. Appl. Math, 70 (1), 21-48). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.
Conjunct with the universal acceleration in information growth, financial services have been immersed in an evolution of information dynamics. It is not just the dramatic increase in volumes of data, but the speed, the complexity and the unpredictability of big-data phenomena that have compounded the challenges faced by researchers and practitioners in financial services. Math, statistics and technology have been leveraged creatively to create analytical solutions. Given the many unique characteristics of financial bid data (FBD) it is necessary to gain insights into strategies and models that can be used to create FBD specific solutions. Behavioral finance data, a subset of FBD, is seeing exponential growth and this presents an unprecedented opportunity to study behavioral finance employing big data analytics methodologies. The present study maps machine learning (ML) techniques and behavioral finance categories to explore the potential for using ML techniques to address behavioral aspects in FBD. The ontological feasibility of such an approach is presented and the primary purpose of this study is propositioned- ML based behavioral models can effectively estimate performance in FBD. A simple machine learning algorithm is successfully employed to study behavioral performance in an artificial stock market to validate the propositions. Keywords: Information; Big Data; Electronic Markets; Analytics; Behavior
This book, which is in Spanish, provides detailed descriptions, including over 550 mathematical formulas, for over 150 trading strategies across a host of asset classes (and trading styles). This includes stocks, options, fixed income, futures, ETFs, indexes, commodities, foreign exchange, convertibles, structured assets, volatility (as an asset class), real estate, distressed assets, cash, cryptocurrencies, miscellany (such as weather, energy, inflation), global macro, infrastructure, and tax arbitrage. Some strategies are based on machine learning algorithms (such as artificial neural networks, Bayes, k-nearest neighbors). We also give: source code for illustrating out-of-sample backtesting with explanatory notes; around 2,000 bibliographic references; and over 900 glossary, acronym and math definitions. The presentation is intended to be descriptive and pedagogical. ----- Este libro proporciona descripciones detalladas, que incluyen m\'as de 550 f\'ormulas matem\'aticas, para m\'as de 150 estrategias de trading para una gran cantidad de clases de activos y estilos de trading. Esto incluye acciones, opciones, bonos (renta fija), futuros, ETFs, \'indices, commodities, divisas, bonos convertibles, activos estructurados, volatilidad (como clase de activos), bienes inmuebles, activos en distress, efectivo, criptomonedas, miscel\'aneos (como clima, energ\'ia, inflaci\'on), macro global, infraestructura y arbitraje impositivo. Algunas estrategias se basan en algoritmos de aprendizaje autom\'atico (como redes neuronales artificiales, Bayes, k vecinos m\'as cercanos). El libro tambi\'en incluye c\'odigo para backtesting fuera de la muestra con notas explicativas; cerca de 2,000 referencias bibliogr\'aficas; m\'as de 900 t\'erminos que comprenden el glosario, acr\'onimos y definiciones matem\'aticas. La presentaci\'on pretende ser descriptiva y pedag\'ogica.
We show that the basic reproduction number of an SIS patch model with standard incidence is either strictly decreasing and strictly convex with respect to the diffusion coefficient of infected subpopulation if the patch reproduction numbers of at least two patches in isolation are distinct or constant otherwise. Biologically, it means that fast diffusion of the infected people reduces the risk of infection. This completely solves and generalizes a conjecture by Allen et al. ({\it SIAM J Appl Math}, 67: 1283-1309, 2007). Furthermore, a substantially improved lower bound on the multipatch reproduction number, a generalized monotone result on the spectral bound the Jacobian matrix of the model system at the disease-free equilibrium, and the limiting endemic equilibrium are obtained. The approach and results can be applied to a class of epidemic patch models where only one class of infected compartments migrate between patches and one transmission route is involved.
Although the scientific digital library is growing at a rapid pace, scholars/students often find reading Science, Technology, Engineering, and Mathematics (STEM) literature daunting, especially for the math-content/formula. In this paper, we propose a novel problem, ``mathematics content understanding'', for cyberlearning and cyberreading. To address this problem, we create a Formula Evolution Map (FEM) offline and implement a novel online learning/reading environment, PDF Reader with Math-Assistant (PRMA), which incorporates innovative math-scaffolding methods. The proposed algorithm/system can auto-characterize student emerging math-information need while reading a paper and enable students to readily explore the formula evolution trajectory in FEM. Based on a math-information need, PRMA utilizes innovative joint embedding, formula evolution mining, and heterogeneous graph mining algorithms to recommend high quality Open Educational Resources (OERs), e.g., video, Wikipedia page, or slides, to help students better understand the math-content in the paper. Evaluation and exit surveys show that the PRMA system and the proposed formula understanding algorithm can effectively assist master and PhD students better understand the complex math-content in the class readings.
In aircraft industry, market needs evolve quickly in a high competitiveness context. This requires adapting a given aircraft model in minimum time considering for example an increase of range or the number of passengers (cf A330 NEO family). The computation of loads and stress to resize the airframe is on the critical path of this aircraft variant definition: this is a consuming and costly process, one of the reason being the high dimen-sionality and the large amount of data. This is why Airbus has invested since a couple of years in Big Data approaches (statistic methods up to machine learning) to improve the speed, the data value extraction and the responsiveness of this process. This paper presents recent advances in this work made in cooperation between Airbus, ENAC and Institut de Math{\'e}-matiques de Toulouse in the framework of a proof of value sprint project. It compares the influence of three dimensional reduction techniques (PCA, polynomial fitting, combined) on the extrapolation capabilities of Regression Trees based algorithms for loads prediction. It shows that AdaBoost with Random Forest offers promising results in average in terms of accuracy and computational time to estimate loads on which a PCA is applied only on the outputs.
We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrarily to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth-death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by N\'u\~nez Rodr\'iguez and Neves (J. Math. Biol. (2016) 73:1665--1690). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.
We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchard, B., Deng, S. and Tan, X., Super-replication with proportional transaction cost under model uncertainty, Math. Fin., 29(3):837-860, 2019] and to show the duality under the more natural condition of No Strict Arbitrage. In addition, we extend the results to models with portfolio constraints.
Digital mathematical libraries (DMLs) such as arXiv, Numdam, and EuDML contain mainly documents from STEM fields, where mathematical formulae are often more important than text for understanding. Conventional information retrieval (IR) systems are unable to represent formulae and they are therefore ill-suited for math information retrieval (MIR). To fill the gap, we have developed, and open-sourced the MIaS MIR system. MIaS is based on the full-text search engine Apache Lucene. On top of text retrieval, MIaS also incorporates a set of tools for preprocessing mathematical formulae. We describe the design of the system and present speed, and quality evaluation results. We show that MIaS is both efficient, and effective, as evidenced by our victory in the NTCIR-11 Math-2 task.
There are no two identical leaves in the world, so how to find effective markers or features to distinguish them is an important issue. Function transformation, such as f(x,y) and f(x,y,z), can transform two, three, or multiple input/observation variables (in biology, it generally refers to the observed/measured value of biomarkers, biological characteristics, or other indicators) into a new output variable (new characteristics or indicators). This provided us a chance to re-cognize objective things or relationships beyond the original measurements. For example, Body Mass Index, which transform weight and high into a new indicator BMI=x/y^2 (where x is weight and y is high), is commonly used in to gauge obesity. Here, we proposed a new system, Funomics (Function Transformation Omics), for understanding the world in a different perspective. Funome can be understood as a set of math functions consist of basic elementary functions (such as power functions and exponential functions) and basic mathematical operations (such as addition, subtraction). By scanning the whole Funome, researchers can identify some special functions (called handsome functions) which can generate the novel important output variable (characteristics or indicators). We also start "the Funome project" to develop novel methods, function library and analysis software for Funome studies. The Funome project will accelerate the discovery of new useful indicators or characteristics, will improve the utilization efficiency of directly measured data, and will enhance our ability to understand the world. The analysis tools and data resources about the Funome project can be found gradually at http://www.funome.com.
We apply techniques in natural language processing, computational linguistics, and machine-learning to investigate papers in hep-th and four related sections of the arXiv: hep-ph, hep-lat, gr-qc, and math-ph. All of the titles of papers in each of these sections, from the inception of the arXiv until the end of 2017, are extracted and treated as a corpus which we use to train the neural network Word2Vec. A comparative study of common n-grams, linear syntactical identities, word cloud and word similarities is carried out. We find notable scientific and sociological differences between the fields. In conjunction with support vector machines, we also show that the syntactic structure of the titles in different sub-fields of high energy and mathematical physics are sufficiently different that a neural network can perform a binary classification of formal versus phenomenological sections with 87.1% accuracy, and can perform a finer five-fold classification across all sections with 65.1% accuracy.
This paper is based on the complete classification of evolutionary scenarios for the Moran process with two strategies given by Taylor et al. (B. Math. Biol. 66(6): 1621--1644, 2004). Their classification is based on whether each strategy is a Nash equilibrium and whether the fixation probability for a single individual of each strategy is larger or smaller than its value for neutral evolution. We improve on this analysis by showing that each evolutionary scenario is characterized by a definite graph shape for the fixation probability function. A second class of results deals with the behavior of the fixation probability when the population size tends to infinity. We develop asymptotic formulae that approximate the fixation probability in this limit and conclude that some of the evolutionary scenarios cannot exist when the population size is large.
Despite simplicity, the synchronous cellular automaton [D.A. Young, Math. Biosci. 72, 51 (1984)] enables reconstructing basic features of patterns of skin. Our extended model allows studying the formatting of patterns and their temporal evolution also on the favourable and hostile environments. As a result, the impact of different types of an environment is accounted for the dynamics of patterns formation. The process is based on two diffusible morphogens, the short-range activator and the long-range inhibitor, produced by differentiated cells (DCs) represented as black pixels. For a neutral environment, the extended model reduces to the original one. However, even the reduced model is statistically sensitive to a type of the initial distribution of DCs. To compare the impact of the uniform random distribution of DCs (R-system) and the non-uniform distribution in the form of random Gaussian-clusters (G-system), we chose inhibitor as the control parameter. To our surprise, in the neutral environment, for the chosen inhibitor-value that ensures stable final patterns, the average size of final G-populations is lower than in the R-case. In turn, when we consider the favourable environment, the relatively bigger shift toward higher final concentrations of DCs appears in the G. Thus, in the suitably favourable environment, this order can be reversed. Furthermore, the different critical values of the control parameter for the R and the G suggest some dissimilarities in temporal evolution of both systems. In particular, within the proper ranges of the critical values, their oscillatory behaviours are different. The respective temporal evolutions are illustrated by a few examples.
In this article, the theory of hierarchical evolutive systems of Ehresmann and Vandremeersch [Bull. Math. Bio. 49, 13-50 (1987)] is improved by considering the categories of the theory as fuzzy sets whose elements are the composite objects formed by the arrows and corresponding vertices of their embedded graphs. This way each category can be represented as a point in the states space [0.1]**N. The introduction of a diffeomorphism, that acts in this context as a functor between categories, allows to define a measure preserving dynamical system. In particular, we apply this formalism to describe a living single cell. We propose for its state at a given time a hirerchical category with three levels (molecular, coarse-grained and cellular levels) related by adequate colimits. Each level involves the main functional and structural modules in which the cell can be partitioned. The time evolution of the cell is drived by a transformation which is a N-dimensional generalization of the Ricker map whose parameters we propose to be determined by requiring that, as hallmark of its behavior, the living cell to evolve at the edge of chaos. From the dynamical point of view this property manifests in the fact that the largest Lyapunov exponent is equal to zero. Since in a rather complete model of the living cell the huge number of involved parameters can make of the calculations a hard task, we also propose a toy model, with fewer parameters to be determined, which emphasizes the cellular fission.
The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: From a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in FACC Chalub and MO Souza, J. Math. Biol. 75: 1735, 2017., which yields a fitness function that realises this fixation pattern in the Wright-Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using $d$-player game theory, provided $d$ is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.
Market timing is an investment technique that tries to continuously switch investment into assets forecast to have better returns. What is the likelihood of having a successful market timing strategy? With an emphasis on modeling simplicity, I calculate the feasible set of market timing portfolios using index mutual fund data for perfectly timed (by hindsight) all or nothing quarterly switching between two asset classes, US stocks and bonds over the time period 1993--2017. The historical optimal timing path of switches is shown to be indistinguishable from a random sequence. The key result is that the probability distribution function of market timing returns is asymetric, that the highest probability outcome for market timing is a below median return. Put another way, simple math says market timing is more likely to lose than to win---even before accounting for costs. The median of the market timing return probability distribution can be directly calculated as a weighted average of the returns of the model assets with the weights given by the fraction of time each asset has a higher return than the other. For the time period of the data the median return was close to, but not identical with, the return of a static 60:40 stock:bond portfolio. These results are illustrated through Monte Carlo sampling of timing paths within the feasible set and by the observed return paths of several market timing mutual funds.
The working memory capacity (WMC) of 400 Russian college students was measured using the Tarnow Unchunkable Test [2] which tests WMC alone without requiring explicit working memory operations. We found small-sized WMC differences by gender and the possibility that the male/female ratio increases for low and high WMC creating a u-shaped curve. The gender proportion in each academic fields was a strong determinant of the average WMC (r2=0.2 for the 3-item test and r2=0.5 for the 4-item test), associating "academic female" (law, history) with holistic thinking and "academic male" with reductive thinking (physics, computer science, math). Within academic fields there were no WMC gender differences. The male/female ratios for the different fields are strongly amplified from the WMC male/female ratios, by factors of 12-14.