A new error bound for linear complementarity problems involving B− B- matrices

In this paper, a new error bound for the linear complementarity problems of $B-$matrices which is a subclass of the $P-$matrices is presented. Theoretical analysis and numerical example illustrate that the new error bound improves some existing results

Impulse Control in Finance: Numerical Methods and Viscosity Solutions

The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

Impulse Control in Finance: Numerical Methods and Viscosity Solutions

The goal of this thesis is to provide efficient and provably convergent numerical methods for solving partial differential equations (PDEs) coming from impulse control problems motivated by finance. Impulses, which are controlled jumps in a stochastic process, are used to model realistic features in financial problems which cannot be captured by ordinary stochastic controls. The dynamic programming equations associated with impulse control problems are Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) Other than in certain special cases, the numerical schemes that come from the discretization of HJBQVIs take the form of complicated nonlinear matrix equations also known as Bellman problems. We prove that a policy iteration algorithm can be used to compute their solutions. In order to do so, we employ the theory of weakly chained diagonally dominant (w.c.d.d.) matrices. As a byproduct of our analysis, we obtain some new results regarding a particular family of Markov decision processes which can be thought of as impulse control problems on a discrete state space and the relationship between w.c.d.d. matrices and M-matrices. Since HJBQVIs are nonlocal PDEs, we are unable to directly use the seminal result of Barles and Souganidis (concerning the convergence of monotone, stable, and consistent numerical schemes to the viscosity solution) to prove the convergence of our schemes. We address this issue by extending the work of Barles and Souganidis to nonlocal PDEs in a manner general enough to apply to HJBQVIs. We apply our schemes to compute the solutions of various classical problems from finance concerning optimal control of the exchange rate, optimal consumption with fixed and proportional transaction costs, and guaranteed minimum withdrawal benefits in variable annuities

Numerical methods and accurate computations with structured matrices

Esta tesis doctoral es un compendio de 11 artículos científicos. El tema principal de la tesis es el Álgebra Lineal Numérica, con énfasis en dos clases de matrices estructuradas: las matrices totalmente positivas y las M-matrices. Para algunas subclases de estas matrices, es posible desarrollar algoritmos para resolver numéricamente varios de los problemas más comunes en álgebra lineal con alta precisión relativa independientemente del número de condición de la matriz. La clave para lograr cálculos precisos está en el uso de una parametrización diferente que represente la estructura especial de la matriz y en el desarrollo de algoritmos adaptados que trabajen con dicha parametrización.Las matrices totalmente positivas no singulares admiten una factorización única como producto de matrices bidiagonales no negativas llamada factorización bidiagonal. Si conocemos esta representación con alta precisión relativa, se puede utilizar para resolver ciertos sistemas de ecuaciones y para calcular la inversa, los valores propios y los valores singulares con alta precisión relativa. Nuestra contribución en este campo ha sido la obtención de la factorización bidiagonal con alta precisión relativa de matrices de colocación de polinomios de Laguerre generalizados, de matrices de colocación de polinomios de Bessel, de clases de matrices que generalizan la matriz de Pascal y de matrices de q-enteros. También hemos estudiado la extensión de varias propiedades óptimas de las matrices de colocación de B-bases normalizadas (que en particular son matrices totalmente positivas). En particular, hemos demostrado propiedades de optimalidad de las matrices de colocación del producto tensorial de B-bases normalizadas.Si conocemos las sumas de filas y las entradas extradiagonales de una M-matriz no singular diagonal dominante con alta precisión relativa, entonces podemos calcular su inversa, determinante y valores singulares también con alta precisión relativa. Hemos buscado nuevos métodos para lograr cálculos precisos con nuevas clases de M-matrices o matrices relacionadas. Hemos propuesto una parametrización para las Z-matrices de Nekrasov con entradas diagonales positivas que puede utilizarse para calcular su inversa y determinante con alta precisión relativa. También hemos estudiado la clase denominada B-matrices, que está muy relacionada con las M-matrices. Hemos obtenido un método para calcular los determinantes de esta clase con alta precisión relativa y otro para calcular los determinantes de las matrices de B-Nekrasov también con alta precisión relativa. Basándonos en la utilización de dos matrices de escalado que hemos introducido, hemos desarrollado nuevas cotas para la norma infinito de la inversa de una matriz de Nekrasov y para el error del problema de complementariedad lineal cuando su matriz asociada es de Nekrasov. También hemos obtenido nuevas cotas para la norma infinito de las inversas de Bpi-matrices, una clase que extiende a las B-matrices, y las hemos utilizado para obtener nuevas cotas del error para el problema de complementariedad lineal cuya matriz asociada es una Bpi-matriz. Algunas clases de matrices han sido generalizadas al caso de mayor dimensión para desarrollar una teoría para tensores extendiendo la conocida para el caso matricial. Por ejemplo, la definición de la clase de las B-matrices ha sido extendida a la clase de B-tensores, dando lugar a un criterio sencillo para identificar una nueva clase de tensores definidos positivos. Hemos propuesto una extensión de la clase de las Bpi-matrices a Bpi-tensores, definiendo así una nueva clase de tensores definidos positivos que puede ser identificada en base a un criterio sencillo basado solo en cálculos que involucran a las entradas del tensor. Finalmente, hemos caracterizado los casos en los que las matrices de Toeplitz tridiagonales son P-matrices y hemos estudiado cuándo pueden ser representadas en términos de una factorización bidiagonal que sirve como parametrización para lograr cálculos con alta precisión relativa.<br /

Numerical Methods for Hamilton-Jacobi-Bellman Equations with Applications

Hamilton-Jacobi-Bellman (HJB) equations are nonlinear controlled partial differential equations (PDEs). In this thesis, we propose various numerical methods for HJB equations arising from three specific applications. First, we study numerical methods for the HJB equation coupled with a Kolmogorov-Fokker-Planck (KFP) equation arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. The main novelty of our approach is that we subtract artificial viscosity from the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. The convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature. Next, we investigate numerical methods for the HJB formulation that arises from the mass transport image registration model. We convert the PDE of the model (a Monge-Ampère equation) to an equivalent HJB equation, propose a monotone mixed discretization, and prove that it is guaranteed to converge to the viscosity solution. Then we propose multigrid methods for the mixed discretization, where we set wide stencil points as coarse grid points, use injection at wide stencil points as the restriction, and achieve a mesh-independent convergence rate. Moreover, we propose a novel periodic boundary condition for the image registration PDE, such that when two images are related by a combination of a translation and a non-rigid deformation, the numerical scheme recovers the underlying transformation correctly. Finally, we propose a deep neural network framework for the HJB equations emerging from the study of American options in high dimensions. We convert the HJB equation to an equivalent Backward Stochastic Differential Equation (BSDE), introduce the least squares residual of the BSDE as the loss function, and propose a new neural network architecture that utilizes the domain knowledge of American options. Our proposed framework yields American option prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer