# Research Lines:

NUMERICAL ANALYSIS AND COMPUTING IN FINANCE

The modern theory of finance and especially the areas of financial derivatives and financial engineering depend strongly on mathematics. Since the subject was kick-started by the work of Black and Scholes and Merton there has been a very fruitful two-way technology transfer between a range of areas of mathematics and the financial applications. While the standard Black Scholes model was the single most important step in the development of modern derivative asset analysis, the underlying assumptions of constant volatility and a perfectly liquid market are clearly at odds with reality. As a consequence a number of approaches have been developed for dealing with the pricing and, in particular, the hedging of derivatives taking into account the issues having influences in the reality of imperfect markets. Thus, several mathematical models have been recently proposed in the context of limited liquidity and taking into account transaction costs. In practice, these models are required to approximate them in a reliable way. Our goal is to construct efficient numerical methods for Financial Nonlinear Model Problems, not only regarding to the computational point of view but also paying attention to the quality of the approximation throughout the careful numerical analysis of the proposed numerical schemes. Careless numerical computations may waste a good mathematical model. The validation and the simulation of the approximating solution will be also considered in our projects.

RESEARCHERS:

Dr. Lucas Antonio Jódar Sánchez (IMM) , e-mail: ljodar@imm.upv.es

Dr. Rafael Company Rossi (IMM) , e-mail: rcompany@imm.upv.es

Dra. María Consuelo Casabán Bartual (IMM) , e-mail: macabar@mat.upv

RANDOM DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS

Deterministic mathematical models based upon differential equations are, to certain point of view, very useful to represent the complex reality in different scientific and technical fields. The formulation of such models requires, in practice, exhaustive measurements of relevant variables in order to evaluate the different parameters of the model. This implies a level of uncertainty, as a consequence of measurements errors, and randomness must be introduced in the formulation of these models from the beginning. Moreover, in realistic scenarios we cannot set the variables to be measured to definite values because these also depends on many factors whose accurate prediction is far from deterministic. For example: the propagation of a biological species depends on its growth rate, which is influenced by a large number of environmental and genetic factors whose complex behaviour cannot be considered deterministic; geological phenomena such as earthquake displacements requires the introduction of spatial randomness into the model in order to describe properly the ground’s heterogeneities. Precise terrain configurations cannot be determined with precision because of its inaccessibility and the unfeasibility of the associated costs. The determination of prices of financial options is a function of the sophisticated conditions on global markets depending on random factors such as the investment confidence, international political risks or the decisions imposed by governments. This random approach, closer to the complexity of reality instead of classical deterministic approaches, makes more reasonable to include randomness into the formulation of mathematical models based upon differential equations which can be useful to scientific applications. By exploiting the so-called mean square and mean fourth stochastic calculus, the main research lines of the group are:

1. Numerical techniques allowing to obtain approximate stochastic processes (in the mean square sense) of the solution of random differential equations (by random multistep methods and other numerical schemes) and also of partial differential diffusion and wave equations (through the application of random difference methods). This includes the search of mean square stability and consistency conditions for every case.

2. Analytic-numerical methods based on a generalization of the Fröbenius method from the deterministic case (majorant method) to the random scenario to build power series approximations to the solution processes of random differential equations with very general second member (non-linear, mean square continuous, etc.).

3. For every approximate stochastic process proposed by the previous methods, we implement techniques to calculate the main statistical information of the models. In particular, we are interested in calculating the mean and the correlation functions.

4. Applications to different fields of interest, such as Epidemiology, Finances, Chemistry, Seismology, Hidrology, etc.

In sharp contrast with the techniques usually employed in order to study stochastic differential equations based upon Itô calculus (where the randomness or noise is usually introduced by means of an ideal stationary and Gaussian noise, the so-called white noise), we also study random differential equations with an uncertainty modelled by a wide variety of regular stochastic processes. In the application stage, we will use models formulated in different scientific and technical fields in order to validate the mathematical methods proposed. On the other hand, recently the group is also interested in the solution of random differential equations by applying the Wiener-Hermite expansion method as well as the so-called polynomial chaos that allow to take advantage of orthogonality properties of certain class of polynomials to represent the solution of random differential equations.

RESEARCHERS:

Dr. Lucas Antonio Jódar Sánchez (IMM) , e-mail: ljodar@imm.upv.es

Dr. Juan Carlos Cortés López (IMM) , e-mail: jccortes@imm.upv.es

Dra. Laura Villafuerte (Universidad Autónoma de Chiapas, Mexico) , e-mail: lva5@hotmail.com

Gema Calbo Sanjuan (IMM) , e-mail: gcalbo@imm.upv.es

MODELLING THE SPREAD OF THE INFECTIOUS DISEASES AND SOCIALLY TRANSMITTED HABITS

Mathematical epidemiology is a multidisciplinary area where doctors and mathematicians work together in order to understand the mechanisms behind the viruses and bacteria spread and transmission among individuals and/or animals. From the celebrated paper of Kermack and McKendrick in 1927, mathematical epidemiology has been helping to answer questions about the transmission dynamics of influenza, AH1N1, malaria, tuberculosis, sexually transmitted diseases (STD), etc.

In our group, we deal with of the study of Respiratory Syncytial Virus (RSV), which affects more acutely to children younger than a year old in such a way that around 1,300 of them have to be hospitalised in the Spanish region of Valencia every year. At this moment, we are working in the study and analysis of effective vaccination strategies to be implemented over the next few years when the vaccine will be available.

Also, we are starting to study the transmission dynamics of Meningococcal C, a bacterium that may produce meningitis, a disease with a worrying death rate, overall in children.

The methods in mathematical epidemiology can also be used when we consider socially transmitted habits, due to recent papers where the authors show that our daily habits may be transmitted to people nearby: parents, siblings, friends, workmates, etc., and vice versa, they may transmit their habits to us. Some examples of habits we dealt with are unhealthy lifestyles leading to gain weight, smoking, cocaine and alcohol abuse and even what we think. To study properly this kind of problems, we collaborate with an endocrinologist and a psychiatrist.

In both cases, spread of infectious diseases and transmission of habits, the general method to be followed is the same:

* To get data related to the problem.

* To build the model. Here, the collaboration with the experts is of great value.

* To estimate parameters.

* To predict the future behaviour of the disease or the habit on the population. Here, the opinion of the experts is also important.

* To simulate strategies (usually changes on the model parameters) to observe their effect over next few years. These strategies use to be vaccination programmes or public health policies.

RESEARCHERS:

Dr. Rafael Jacinto Villanueva Micó (IMM) , e-mail: rjvillan@imm.upv.es

Dr. Lucas Antonio Jódar Sánchez (IMM) , e-mail: ljodar@imm.upv.es

Dr. Gregorio Rubio Navarro(IMM) , e-mail: grubio@mat.upv.es

Dr. Luis Acedo Rodríguez (IMM) , e-mail: luiacrod@imm.upv.es

Dr. José Antonio Moraño Fernández (IMM) , e-mail: jomofer@mat.upv.es

Dr. Francisco Santonja (Universidad de Valencia, Spain) , e-mail: francisco.santonja@uv.es

TECHNICAL MANAGEMENT OF WATER DISTRIBUTION SYSTEMS (GROUP FLUING):

* Water leakage-Detection by non-destructive techniques (ground penetrating radar).

* Water leakage-Economic evaluation and recovery programs.

* WDS vulnerability analysis.

* Analysis models for WDS.

* Hydraulic and energetic efficiency.

* Optimal design of WDS by heuristic and evolutionary methods.

* Hydraulic transients (water-hammer) and protection system design.

* Design and planning of District Mettered Areas (DMA).

* Intelligent Data Analysis and Data Mining Applications.

* Rehabilitation and maintenance models.

FLUID SYSTEMS MODELLING: PHYSICAL AND MATHEMATICAL MODELLING

* Software.

* Physical models.

RESEARCHERS:

Dr. Joaquín Izquierdo Sebastián (IMM) , e-mail: jizquier@gmmf.upv.es

Dr. Rafael Pérez García (IMM) , e-mail: rperez@gmmf.upv.es

Carlos Daniel Alonso Guzmán (IMM, Ph. D. student) ,

David Ayala Cabrera (IMM, Ph. D. Student) , e-mail: daaycab@posgrado.upv.es

Xitlali Virginia Delgado Galván (IMM, Ph. D. Student) , e-mail: xitdelga@doctor.upv.es

Joanna Alicia Gutiérrez Pérez (IMM, Ph. D. Student) ,

Manuel Herrera Fernández (IMM, Ph. D. Student) , e-mail: mahefe@upvnet.upv.es

Eva Ramos Martínez (IMM, Ph. D. Student) , e-mail: evarama@upvnet.upv.es

Francisco Javier Izquierdo Sebastián (ASIC) , e-mail: ico@cc.upv.es

Webpage of the Fluing Group

Blog of the Fluing Group

MATRIX ANALYSIS IN CONTROL SYSTEMS

This group is integrated by 12 researchers. During the next 3 years we are interested to study of properties of matrices with application to the mathematical theory of control systems for modeling real processes (biological, economic, biomedical and agronomic). In this study, it is necessary to know more on the underlying matrix theory, which are projection matrices, generalized inverses, involutive matrices, the Jordan structure and the singular value decomposition of singular matrices and factorizations of totally positive matrices. This matrix information will be used to obtain conditions for the structured control system parameters are identifiable and to construct algorithms that determine these variables. Structured systems may be subject to disturbance, therefore the stability of parameter-dependent systems where you can analyze the type of disturbances and their range of variation is considered in this project. Applying mathematical models to control problems of agronomy is one of the objectives, as well as characterization of quasi-positive systems and their applications, in order to the theory derived above can be used in real systems. Finally, the study of bidimensional control systems (2-D) is another objective, concretely, we will focus on the local reachability index of these systems.

1. Matrix Analysis in control positive systems, use permutation and involutive matrices.

2. Jordan form of square singular matrices, Singular value decomposition of rectangular singular matrices and applications.

3. Identifiability property for the structured control systems.

4. Stability of parameter-dependent systems.

5. Modelization of control problems in Agronomy and characterization of quasi-positive systems and their applications.

6. Local reachability index of bidimensional control systems (2-D).

7. Generalized inverses: theoretical and applied approaches.

RESEARCHERS:

Dr. Rafael Cantó Colomina (IMM), e-mail: rcanto@imm.upv.es

Dr. Néstor Javier Thome Coppo (IMM), e-mail: njthome@mat.upv.es

Dr. Ana María Urbano Salvador (IMM), e-mail: amurbano@imm.upv.es

Dr. Peregrina del Carmen Coll Aliaga (IMM), e-mail: mccoll@imm.upv.es

Dr. Elena Sánchez Juan (IMM), e-mail: esanchezj@imm.upv.es

ANALYSIS OF SOCIAL NETWORK SITES

To administrate a Social Network Site it is very important to know the structure of the network and its dynamical behavior. In particular, the administrators usually want to know the users that are important, popular or leaders attending to some features. It is also important to give advice to the users of the network to enhance the linkage of the social community. We focus on the analysis of Internet Search Engines such as Google and develop numerical models of the Internet Network.

RESEARCHERS:

Dr. Francisco Pedroche Sánchez (IMM) , e-mail: pedroche@imm.upv.es

ITERATIVE METHODS FOR SOLVING LINEAR AND NON-LINEAR SYSTEMS.

Iterative methods are designed to approximate the solution of linear and non-linear system of equations. This is an important area in Numerical Analysis and Applied Sciences in general because many problems in Engineering or Science involve the solution of equations of this kind. For example in the discretization techniques for integral equations, partial differential equations, orbit determination algorithms, dynamical models for chemical reactors, radioactive transfer problems, global positioning systems (GPS), among others…

In the last years many articles have been published related to the study and applications of iterative methods. This broad literature reveals that the design and analysis of these methods is a dynamical and live branch in Numerical Analysis with may promising and interesting applications.

Our research team is working in the following specific issues, among others:

**Multi-step iterative methods**. Design, analysis and study of the convergence of new multi-step methos with or without memory.

**Steffensen methods**. Design and study of iterative methods free from derivatives, these are particularly interesting for functions which are not soft.

**Iterative methods for singular problems. **These methods can be applied to multidimensional problems, in which the jacobian matrix is singular or ill-conditioned.

**Interval Methods**. These methods are applied to equations or system of equations.

**Iterative methods in Banach space**. These are an extension of iterative methods for equations and operator systems in Banach spaces by improving Kantarovich convergence conditions.

**Dynamical study for iterative methods**. From the point of view of a complex dynamics, the behaviour of the rational function associated with an iterative method provides us with a valuable information about its stability and efficiency. The study of the parameter space, dynamical plane, Fatou’s and Julia’sets …for every method or family of methods is one of the task tackled in this studies.

**Application to engineering problems**. The methods are designed, adapted, implemented and carried out in order to solve real engineering problems. Among others, we should cite electromagnetic theory problems, non-linear waves, optimization or image signal processing.

RESEARCHERS:

Dr. Juan R. Torregrosa Sánchez (IMM), e-mail: jrtorre@mat.upv.es

Dra. Alicia Cordero Barbero (IMM), e-mail: acordero@mat.upv.es

Dra. Eulalia Martínez Molada (IMPA), e-mail: eumarti@mat.upv.es

Dr. José Luis Hueso Pagoaga (IMM), e-mail: jlhueso@mat.upv.es

Dra. Pura Vindel Cañas (UJI), e-mail: vindel@mat.uji.es

D. Manuel Abad Rodríguez (IMM), e-mail: maabrod@mat.upv.es

D. Francisco Chicharro López (IMM, Ph.D. Student), e-mail: frachilo@teleco.upv.es