Stochastic Differential Equations and Applications

RESUMEN: Esta memoria recoge un estudio de las ecuaciones diferenciales estocásticas. Estas ecuaciones son importantes ya que modelan fenómenos inestables debido a la aleatoriedad de algunas de sus componentes. En el primer capítulo se estudia el movimiento browniano como proceso estocástico básico, su construcción y propiedades. En el segundo capítulo se desarrolla la teoría de integración estocástica de Itô y el Teorema de existencia y unicidad de solución. Se trata también la fórmula de Itô y ejemplos clásicos de ecuaciones diferenciales estocásticas. Finalmente, en el último capítulo se recogen aplicaciones numéricas orientadas a diferentes campos de estudio como las finanzas, la farmacología o la epidemiología.ABSTRACT: This report is focused on the study of stochastic differential equations. These equations model unstable phenomena due to the randomness of some of their components. The first chapter shows the Brownian motion as a basic stochastic process, its construction and properties. In the second chapter Itô's stochastic integration theory and existence and uniqueness Theorem are studied. Itô's formula and classic examples of stochastic di_erential equations are also studied. Finally, last chapter shows numerical applications to several fields such as finances, pharmacology or epidemiology.Grado en Matemática

Weather-type-conditioned calibration of Tropical Rainfall Measuring Mission precipitation over the South Pacific Convergence Zone

The South Pacific region is an area affected by characteristic precipitation patterns undergoing extreme events such as tropical cyclones and droughts. First, a daily weather typing of precipitation is presented, based on principal component analysis and k-means clustering using precipitation and atmospheric circulation variables derived from sea-level pressure and wind reanalysis fields. As a result, five weather types (WTs) are presented, able to capture distinct precipitation spatiotemporal patterns, interpretable in terms of salient regional climate features. Second, we undertake the calibration of the TRMM precipitation product using a set of rain gauge stations as reference and scaling and empirical quantile mapping (eQM) as calibration techniques. Furthermore, we build upon the weather-type classification to compare the results with a WTconditioned calibration approach. Overall, our results underpin the need of adjusting the existing TRMM biases, mostly relevant for the upper tail of their distribution, and advocate the use of correction techniques able to deal with quantile-dependent biases-such as eQM-instead of a simple scaling, in order to obtain a more realistic representation of extreme precipitation events. The conditioning has shown only a marginal added value over the simple approach, although this minor improvement may prove relevant for applications focused on extreme event analysis. Furthermore, the weather types created can be applied to a wide variety of conditioned analyses in this region.AFRICULTURES, Grant/Award Number: 774652; Beach4Cast, Grant/Award Number: PID2019-107053RB-I00; CORDyS, Grant/Award Number: PID2020-116595RB-I00; INDECIS, Grant/Award Number: 69046

Modeling tropical cyclone precipitation from satellite data

RESUMEN: En esta memoria se estudia el modelado de la precipitación provocada por la acción de un ciclón tropical. Se busca estimar la cantidad de precipitación en una zona determinada en función de variables atmosféricas, geográficas u oceánicas. Para ello, se han utilizado tres bases de datos principales: precipitaciones medidas a través de satélite, las trazas históricas de ciclones tropicales y la temperatura superficial del mar. Para la simulación del modelo, se genera una circunferencia que tiene como origen el centro del ciclón, con un radio de 500 kilómetros, determinando la zona de actuación del ciclón. Esta representación estará divida en pequeñas celdas, donde en cada una de ellas se estimará la precipitación que cae mediante un ajuste estadístico. El ajuste se apoya en la técnica de SOM, que es una variación de una red neuronal competitiva. Esta herramienta ayuda en la representación de los datos y así facilita proponer el modelo estadístico. El modelo de precipitación está basado en un ajuste estadístico a partir de una distribución mixta. Dicha distribución se compone de una distribución de Bernouilli con parámetro p (variable discreta), y una distribución exponencial de parámetro μ (variable continua).ABSTRACT: This report studies the modeling of precipitation caused by the action of a tropical cyclone. The aim is to estimate the amount of precipitation in a certain area according to atmospheric, geographical or oceanic variables. For this purpose, three main databases have been used: precipitation measured by satellite, historical tracks of tropical cyclones and sea surface temperature. For the simulation of the model, a circumference is generated that has as origin the center of the cyclone, with a radius of 500 kilometers, determining the zone of performance of the cyclone. This representation will be divided in small cells, where in each one of them the precipitation that falls by means of a statistical adjustment will be estimated. The adjustment is based on the SOM technique, which is a variation of a competitive neural network. This tool helps in the representation of the data and thus facilitates the proposal of the statistical model. The precipitation model is based on a statistical adjustment from a mixed distribution. This distribution is composed of a Bernouilli distribution with parameter p (discrete variable), and an exponential distribution with parameter μ (continuous variable).Máster en Ciencia de Dato