Maria C. Mariani

Solving differential equations with unsupervised neural networks

Abstract A recent method for solving differential equations using feedforward neural networks was applied to a non-steady fixed bed non-catalytic solid–gas reactor. As neural networks have universal approximation capabilities, it is possible to postulate them as solutions for a given DE problem that defines an unsupervised error. The training was performed using genetic algorithms and the gradient descent method. The solution was found with uniform accuracy (MSE ∼10−9) and the trained neural network provides a compact expression for the analytical solution over the entire finite domain. The problem was also solved with a traditional numerical method. In this case, solution is known only over a discrete grid of points and its computational complexity grows rapidly with the size of the grid. Although solutions in both cases are identical, the neural networks approach to the DE problem is qualitatively better since, once the network is trained, it allows instantaneous evaluation of solution at any desired number of points spending negligible computing time and memory.

Mountain pass solutions to equations of p-Laplacian type

Abstract This work is devoted to study the existence of solutions to equations of p-Laplacian type. We prove the existence of at least one solution, and under further assumptions, the existence of infinitely many solutions. In order to apply mountain pass results, we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm.

LOCAL EXISTENCE OF SOLUTIONS TO THE TRANSIENT QUANTUM HYDRODYNAMIC EQUATIONS

The existence of weak solutions locally in time to the quantum hydrodynamic equations in bounded domains is shown. These Madelung-type equations consist of the Euler equations, including the quantum Bohm potential term, for the particle density and the particle current density and are coupled to the Poisson equation for the electrostatic potential. This model has been used in the modeling of quantum semiconductors and superfluids. The proof of the existence result is based on a formulation of the problem as a nonlinear Schrodinger–Poisson system and uses semigroup theory and fixed-point techniques.

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.

Solutions to a stationary nonlinear Black-Scholes type equation

We study by topological methods a nonlinear differential equation generalizing the Black–Scholes formula for an option pricing model with stochastic volatility. We prove the existence of at least a solution of the stationary Dirichlet problem applying an upper and lower solutions method. Moreover, we construct a solution by an iterative procedure.

The effects of the Asian crisis of 1997 on emergent markets through a critical phenomena model

In the present work we have analyzed the financial Asian crisis of 1997, and its consequences on emerging markets. We have done so by means of a phase transition model originally presented by A. Johansen and D. Sornette [1]. We have analyzed the crashes on leading indices of Hong Kong (HSI), Turkey (XU100), Mexico (MMX), Brazil (Bovespa) and Argentina (Merval). With the exception of Argentinas index, we were able to obtain optimum values for the critical date, corresponding to the most probable date of the crash.

NUMERICAL SCHEMES FOR OPTION PRICING IN REGIME-SWITCHING JUMP DIFFUSION MODELS

In this paper, we present algorithms to solve a complex system of partial integro-differential equations (PIDEs) of parabolic type. The system is motivated by applications in finance where the solution of the system gives the price of European options in a regime-switching jump diffusion model. The new algorithms are based on theoretical analysis in Florescu et al. (2012) where the proof of convergence of the algorithms is carried out. The problems are also solved using a more traditional approach, where the integral terms (but not the derivative terms) are treated explicitly. Another contribution of this work details a novel type of jump distribution. Empirical evidence suggests that this type of distribution may be more appropriate to model jumps as it makes them more clearly distinguishable from the signal variability.

A Dirichlet problem for an H-system with variable H

We give conditions onH, a continuous and bounded real function inR3, to obtain at least two solutions for the problem (Dir) below.H can be far from being constant in the sense of [9]. Our motivation is a better understanding of the Plateau problem for the prescribed mean curvature equation.

Levy models and long correlations applied to the study of exchange traded funds

This work is devoted to the study of statistical properties of Exchange Traded Funds (ETF). Some of the leading ETF in the market are analysed by using the Hurst and DFA methods to detect long range correlations, and the Levy models to describe the return distributions. It is concluded that the statistical behaviour of the ETF is very similar to the behaviour of the corresponding financial indices that they mimic.

Stationary solutions for two nonlinear Black-Scholes type equations

We study by topological methods two different problems arising in the Black-Scholes model for option pricing. More specifically, we consider a nonlinear differential equation which generalizes the Black-Scholes formula when the volatility is assumed to be stochastic. On the other hand, we study a model with transaction costs.