arXiv subjects
Publications and source records attributed to Beatriz Salvador.
We consider a one-dimensional exclusion dynamics in mild contact with boundary reservoirs. In the diffusive scale, the particles' density evolves as the solution of the heat equation with non-linear Robin boundary conditions. For appropriate choices of the boundary rates, these partial differential equations have more than one stationary solution. We prove the dynamical large deviations principle.
We analyze the generalized symmetric exclusion process, which allows at most $α$ particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter $θ\in\mathbb R$. We prove that the hydrodynamic behavior is given by the heat equation and depending on the value of $θ$, the equation is supplemented with different boundary conditions. Setting $α= 1$ we find the results known in [1] for the symmetric simple exclusion process.
Artificial neural networks (ANNs) have recently also been applied to solve partial differential equations (PDEs). In this work, the classical problem of pricing European and American financial options, based on the corresponding PDE formulations, is studied. Instead of using numerical techniques based on finite element or difference methods, we address the problem using ANNs in the context of unsupervised learning. As a result, the ANN learns the option values for all possible underlying stock values at future time points, based on the minimization of a suitable loss function. For the European option, we solve the linear Black-Scholes equation, whereas for the American option, we solve the linear complementarity problem formulation. Two-asset exotic option values are also computed, since ANNs enable the accurate valuation of high-dimensional options. The resulting errors of the ANN approach are assessed by comparing to the analytic option values or to numerical reference solutions (for American options, computed by finite elements).