Wavelet Bases Made of Piecewise Polynomial Functions: Theory and Applications

We present wavelet bases made of piecewise (low degree) polynomial functions with an (arbitrary) assigned number of vanishing moments. We study some of the properties of these wavelet bases;in particular we consider their use in the approximation of functions and in numerical quadrature. We focus on two applications: integral kernel sparsification and digital image compression and reconstruction. In these application areas the use of these wavelet bases gives very satisfactory results.

A Trading Execution Model Based on Mean Field Games and Optimal Control

We present a trading execution model that describes the behaviour of a big trader and of a multitude of retail traders operating on the shares of a risky asset. The retail traders are modeled as a population of “conservative” investors that: 1) behave in a similar way, 2) try to avoid abrupt changes in their trading strategies, 3) want to limit the risk due to the fact of having open positions on the asset shares, 4) in the long run want to have a given position on the asset shares. The big trader wants to maximize the revenue resulting from the action of buying or selling a (large) block of asset shares in a given time interval. The behaviour of the retail traders and of the big trader is modeled using respectively a mean field game model and an optimal control problem. These models are coupled by the asset share price dynamic equation. The trading execution strategy adopted by the retail traders is obtained solving the mean field game model. This strategy is used to formulate the optimal control problem that determines the behaviour of the big trader. The previous mathematical models are solved using the dynamic programming principle. In some special cases explicit solutions of the previous models are found. An extensive numerical study of the trading execution model proposed is presented. The interested reader is referred to the website: http://www.econ.univpm.it/recchioni/finance/w19 to find material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website:?http://www.econ.univpm.it/recchioni/finance.

The Barone-Adesi Whaley Formula to Price American Options Revisited

This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.

A Video Game Based on Optimal Control and Elementary Statistics

The video game presented in this paper is a prey-predator game where two preys (human players) must avoid three predators (automated players) and must reach a location in the game field (the computer screen) called preys’ home. The game is a sequence of matches and the human players (preys) must cooperate in order to achieve the best perform- ance against their opponents (predators). The goal of the predators is to capture the preys, which are the predators try to have a “rendez vous” with the preys, using a small amount of the “resources” available to them. The score of the game is assigned following a set of rules to the prey team, not to the individual prey. In some situations the rules imply that to achieve the best score it is convenient for the prey team to sacrifice one of his components. The video game pursues two main purposes. The first one is to show how the closed loop solution of an optimal control problem and elementary sta- tistics can be used to generate (game) actors whose movements satisfy the laws of classical mechanics and whose be- haviour simulates a simple form of intelligence. The second one is “educational”, in fact the human players in order to be successful in the game must understand the restrictions to their movements posed by the laws of classical mechanics and must cooperate between themselves. The video game has been developed having in mind as players for children aged between five and thirteen years. These children playing the video game acquire an intuitive understanding of the basic laws of classical mechanics (Newton’s dynamical principle) and enjoy cooperating with their teammate. The video game has been experimented on a sample of a few dozen children. The children aged between five and eight years find the game amusing and after playing a few matches develop an intuitive understanding of the laws of classical me- chanics. They are able to cooperate in making fruitful decisions based on the positions of the preys (themselves), of the predators (their opponents) and on the physical limitations to the movements of the game actors. The interest in the game decreases when the age of the players increases. The game is too simple to interest a teenager. The game engine consists in the solution of an assignment problem, in the closed loop solution of an optimal control problem and in the adaptive choice of some parameters. At the beginning of each match, and when necessary during a match, an assign- ment problem is solved, that is the game engine chooses how to assign to the predators the preys to chase. The resulting assignment implies some cooperation among the predators and defines the optimal control problem used to compute the strategies of the predators during the match that follows. These strategies are determined as the closed loop solution of the optimal control problem considered and can be thought as a (first) form of artificial intelligence (AI) of the preda- tors. In the optimal control problem the preys and the predators are represented as point masses moving according to Newton’s dynamical principle under the action of friction forces and of active forces. The equations of motion of these point masses are the constraints of the control problem and are expressed through differential equations. The formula- tion of the decision process through optimal control and Newton’s dynamical principle allows us to develop a game where the effectiveness and the goals of the automated players can be changed during the game in an intuitive way sim- ply modifying the values of some parameters (i.e. mass, friction coefficient, ...). In a sequence of game matches the predators (automated players) have “personalities” that try to simulate human behaviour. The predator personalities are determined making an elementary statistical analysis of the points scored by the preys in the game matches played and consist in the adaptive choice of the value of a parameter (the mass) that appears in the differential equations that define the movements of the predators. The values taken by this parameter determine the behaviour of the predators and their effectiveness in chasing the preys. The predators personalities are a (second) form of AI based on elementary statistics that goes beyond the intelligence used to chase the preys in a match. In a sequence of matches the predators using this second form of AI adapt their behaviour to the preys’ behaviour. The video game can be downloaded from the website: http://www.ceri.uniroma1.it/ceri/zirilli/w10/.

Homogeneous and Heterogeneous Traffic of Data Packets on Complex Networks: The Traffic Congestion Phenomenon

We study the congestion phenomenon in a mathematical model of the data packets traffic in transmission networks as a function of the topology and of the load of the network. Two types of traffic are considered: homogeneous and heterogeneous traffic. The congestion phenomenon is studied in stationary conditions through the behaviour of two quantities: the mean travel time of a packet and the mean number of packets that have not reached their destination and are traveling in the network. We define a transformation that maps a network having the small world property (Inet 3037 in our numerical experiments) into a (modified) lattice network that has the same number of nodes. This map changes the capacity of the branches of the graphs representing the networks and can be regarded as an “interpolation” between the two classes of networks. Using this transformation we compare the behaviour of Inet 3037 to the behaviour of a modified rectangular lattice and we study the behaviour of the interpolating networks. This study suggests how to change the network topology and the branch capacities in order to alleviate the congestion phenomenon. In the website: http://www.ceri.uniroma1.it/ceri/zirilli/w6 some auxiliary material including animations and stereo?graphic scenes that helps the understanding of this paper is shown.

The SABR Model: Explicit Formulae of the Moments of the Forward Prices/Rates Variable and Series Expansions of the Transition Probability Density and of the Option Prices

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

A Video Game Based on Elementary Differential Equations

In this paper a prey-predator video game is presented. In the video game two predators chase a prey that tries to avoid the capture by the predators and to reach a location in space (i.e. its “home”). The prey is animated by a human player (using a joypad), the predators are automated players whose behaviour is decided by the video game engine. The purpose of the video game is to show how to use mathematical models to build a simple prey-predator dynamics representing a physical system where the movements of the game actors satisfy Newton’s dynamical principle and the behaviour of the automated players simulates a simple form of intelligence. The game is based on a simple set of ordinary differential equations. These differential equations are used in classical mechanics to describe the dynamics of a set of point masses subject to a force chosen by the human player, elastic forces and friction forces (i.e. viscous damping). The software that implements the video game is written in C++ and Delphi. The video game can be downloaded from: http://www.ceri.uniroma1.

Some Explicit Formulae for the Hull and White Stochastic Volatility Model

An explicit formula for the transition probability density function of the Hull and White stochastic volatility model in presence of nonzero correlation between the stochastic differentials of the Wiener processes on the right hand side of the model equations is presented. This formula gives the transition probability density function as a two dimensional integral of an explicitly known integrand. Previously an explicit formula for this probability density function was known only in the case of zero correlation. In the case of nonzero correlation from the formula for the transition probability density function we deduce formulae (expressed by integrals) for the price of European call and put options and closed form formulae (that do not involve integrals) for the moments of the asset price logarithm. These formulae are based on recent results on the Whittaker functions [1] and generalize similar formulae for the SABR and multiscale SABR models [2]. Using the option pricing formulae derived and the least squares method a calibration problem for the Hull and White model is formulated and solved numerically. The calibration problem uses as data a set of option prices. Experiments with real data are presented. The real data studied are those belonging to a time series of the USA S&P 500 index and of the prices of its European call and put options. The quality of the model and of the calibration procedure is established comparing the forecast option prices obtained using the calibrated model with the option prices actually observed in the financial market. The website: http://www.econ.univpm.it/recchioni/finance/w17 contains some auxiliary material including animations and interactive applications that helps the understanding of this paper. More general references to the work of the authors and of their coauthors in mathematical finance are available in the website: http://www.econ.univpm.it/recchioni/finance.

Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

We study two calibration problems for the lognormal SABR model using the moment method and some new formulae for the moments of the logarithm of the forward prices/rates variable. The lognormal SABR model is a special case of the SABR model [1]. The acronym “SABR” means “Stochastic-αβρ” and comes from the original names of the model parameters (i.e., α,β,ρ) [1]. The SABR model is a system of two stochastic differential equations widely used in mathematical finance whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The lognormal SABR model corresponds to the choice β = 1 and depends on three quantities: the parameters??α,ρ and the initial stochastic volatility. In fact the initial stochastic volatility cannot be observed and can be regarded as a parameter. A calibration problem is an inverse problem that consists in determineing the values of these three parameters starting from a set of data. We consider two different sets of data, that is: i) the set of the forward prices/rates observed at a given time on multiple independent trajectories of the lognormal SABR model, ii) the set of the forward prices/rates observed on a discrete set of known time values along a single trajectory of the lognormal SABR model. The calibration problems corresponding to these two sets of data are formulated as constrained nonlinear least-squares problems and are solved numerically. The formulation of these nonlinear least-squares problems is based on some new formulae for the moments of the logarithm of the forward prices/rates. Note that in the financial markets the first set of data considered is hardly available while the second set of data is of common use and corresponds simply to the time series of the observed forward prices/rates. As a consequence the first calibration problem although realistic in several contexts of science and engineering is of limited interest in finance while the second calibration problem is of practical use in finance (and elsewhere). The formulation of these calibration problems and the methods used to solve them are tested on synthetic and on real data. The real data studied are the data belonging to a time series of exchange rates between currencies (euro/U.S. dollar exchange rates).

The Calibration of Some Stochastic Volatility Models Used in Mathematical Finance

Stochastic volatility models are used in mathematical finance to describe the dynamics of asset prices. In these models, the asset price is modeled as a stochastic process depending on time implicitly defined by a stochastic differential Equation. The volatility of the asset price itself is modeled as a stochastic process depending on time whose dynamics is described by a stochastic differential Equation. The stochastic differential Equations for the asset price and for the volatility are coupled and together with the necessary initial conditions and correlation assumptions constitute the model. Note that the stochastic volatility is not observable in the financial markets. In order to use these models, for example, to evaluate prices of derivatives on the asset or to forecast asset prices, it is necessary to calibrate them. That is, it is necessary to estimate starting from a set of data the values of the initial volatility and of the unknown parameters that appear in the asset price/volatility dynamic Equations. These data usually are observations of the asset prices and/or of the prices of derivatives on the asset at some known times. We analyze some stochastic volatility models summarizing merits and weaknesses of each of them. We point out that these models are examples of stochastic state space models and present the main techniques used to calibrate them. A calibration problem for the Heston model is solved using the maximum likelihood method. Some numerical experiments about the calibration of the Heston model involving synthetic and real data are presented.

Acoustic Scattering Cross Sections of Smart Obstacles: A Case Study

Acoustic scattering cross sections of smart furtive obstacles are studied and discussed.A smart furtive obstacle is an obstacle that,when hit by an incoming field,avoids detection through the use of a pressure current acting on its boundary.A highly parallelizable algorithm for computing the acoustic scattering cross section of smart obstacles is developed.As a case study,this algorithm is applied to the(acoustic)scattering cross section of a"smart"(furtive)simplified version of the NASA space shuttle when hit by incoming time-harmonic plane waves,the wavelengths of which are small compared to the characteristic dimensions of the shuttle.The solution to this numerically challenging scattering problem requires the solution of systems of linear equations with many unknowns and equations.Due to the sparsity of these systems of equations,they can be stored and solved using affordable computing resources.A cross section analysis of the simplified NASA space shuttle highlights three findings:i)the smart furtive obstacle reduces the magnitude of its cross section compared to the cross section of a corresponding"passive"obstacle;ii)several wave propagation directions fail to satisfactorily respond to the smart strategy of the obstacle;iii)satisfactory furtive effects along all directions may only be obtained by using a pressure current of considerable magnitude.Numerical experiments and virtual reality applications can be found at the website:http://www.ceri.uniroma1.it/ceri/zirilli/w7.

High Performance Algorithms Based on a New Wavelet Expansion for Time Dependent Acoustic Obstacle Scattering

This paper presents a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems.The method proposed is a generalization of the“operator expansion method”developed by Recchioni and Zirilli[SIAM J.Sci.Comput.,25(2003),1158-1186].The numerical method proposed reduces,via a perturbative approach,the solution of the scattering problem to the solution of a sequence of systems of first kind integral equations.The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and small wavelengths are solved.A computational method has been developed to solve these challenging problems with affordable computing resources.To this aim a new way of using the wavelet transform and new bases of wavelets are introduced,and a version of the operator expansion method is developed that constructs directly element by element in a fully parallelizable way.Several numerical experiments involving realistic obstacles and“small”wavelengths are proposed and high dimensional vector spaces are used in the numerical experiments.To evaluate the performance of the proposed algorithm on parallel computing facilities,appropriate speed up factors are introduced and evaluated.