One-compartment stochastic pharmacokinetic model Ricardo Cano Macias, José Alfredo Jiménez Moscoso, Jorge Mauricio Ruiz Vera Universitas Scientiarum, 2023 In this work, we consider a pharmacokinetic (PK) model with first-order drug absorption and first-order elimination that represent the concentration of drugs in the body, including both the absorption and elimination parts, and we also add a random factor to describe the variability between patients and the environment. Using Itô’s lemma and the Laplace transform, we obtain the solutions in integral form for a single and constant dosage regimen in time. Moreover, formulas for the expected value and the variance for each case of study are presented, which allows the statistical assessment of the proposed models, as well as predicting the ideal path of drug concentration and its uncertainty. These results are important in the long-term analysis of drug concentration and the persistence of therapeutic level. Further, a numerical method for the solution of the stochastic differential equation (SDE) is introducedand developed.
Optimal portfolio selection based on first and second order Markov chains Juan Manuel Gómez Romero, José Alfredo Jiménez Moscoso Lecturas De Economia, 2020 En búsqueda de generar estrategias de inversión en pro de maximizar el rendimiento esperado y minimizar el riesgo, se estudian dos modelos de selección de portafolios óptimos. El primero se ajusta usando rendimientos logarítmicos, y en el segundo se emplea análisis de componentes principales (ACP) a estos rendimientos. Luego, para cada uno de ellos se establece su rendimiento ponderado y se crean unas medidas para establecer los estados de las cadenas de Markov de primer y segundo orden. Esto permite pronosticar si los portafolios conformados tendrán comportamientos alcistas o bajistas dadas las probabilidades de los estados de las cadenas de Markov. Se realiza una aplicación usando los retornos de precios de cierre diarios de 21 acciones del COLCAP, para el periodo comprendido desde enero de 2014 a octubre de 2017. Se concluye que en el mercado colombiano un portafolio conformado bajo ACP de los rendimientos tiene una mayor rentabilidad esperada y un menor riesgo a largo plazo, teniendo una precisión de pronóstico del modelo dados los vectores estacionarios de las cadenas de Markov.
Evaluating operational risk by an inhomogeneous counting process based on Panjer recursion Viswanathan Arunachalam, José Alfredo Jiménez Journal of Operational Risk, 2016 Operational risk (OpRisk) is increasingly being considered an important financial risk. In recent years, it has been given more consideration due to economically disturbing events. The loss distribution approach (LDA) is one of the demanding models suggested by the Basel Committee on Banking Supervision (BCBS). The purpose of this paper is to propose a new approach for determining operational value-at-risk (OpVaR) using an inhomogeneous counting process based on Panjer recursion as the frequency distribution, and generalized Pareto distributions and generalized extreme value distributions are used to model the severities. We focus on the inhomogeneous Panjer process (IPP) and investigate its properties. In this paper, we present the LDA for computing OpRisk using IPP. The closed-form expression for the moment generating function for determining the aggregate loss distribution is derived. We generalize well-known classical models and derive the statistical characteristics for modeling loss distribution. An illustration is presented to demonstrate the applicability of the proposed model in OpRisk and also discuss various special cases of the model.
A Mixture of Generalized Tukey's g Distributions José Alfredo Jiménez, Viswanathan Arunachalam Journal of Probability and Statistics, 2016 Mixtures of symmetric distributions, in particular normal mixtures as a tool in statistical modeling, have been widely studied. In recent years, mixtures of asymmetric distributions have emerged as a top contender for analyzing statistical data. Tukey’sgfamily of generalized distributions depend on the parameters, namely,g, which controls the skewness. This paper presents the probability density function (pdf) associated with a mixture of Tukey’sgfamily of generalized distributions. The mixture of this class of skewed distributions is a generalization of Tukey’sgfamily of distributions. In this paper, we calculate a closed form expression for the density and distribution of the mixture of two Tukey’sgfamilies of generalized distributions, which allows us to easily compute probabilities, moments, and related measures. This class of distributions contains the mixture of Log-symmetric distributions as a special case.
OPtion Pricing Based on A Log-Skew-Normal Mixture J. A. JIMÉNEZ, V. ARUNACHALAM, G. M. SERNA International Journal of Theoretical and Applied Finance, 2015 This paper presents a method for approximating the underlying stock’s distribution by using a Log–Skew–Normal mixture distribution. The basic properties of a mixture of Skew–Normal distributions are reviewed in this paper. We provide a formula for the European option price by assuming that the log price follows a Skew–Normal mixture distribution. We also calculate the “Greeks”, such as delta, gamma and vega. We compare the proposed model with other existing models and consider an example of calibration to real market option data.
Valuation for european derivatives with mixture-Weibull distributions Andrés Mauricio Molina, José Alfredo Jiménez Cuadernos De Economia Colombia, 2015 El modelo Black-Scholes para valoración de opciones europeas se usa bastante en el mercado por su fácil ejecución. Sin embargo, empieza a ser poco preciso en diferentes activos cuya dinámica no es de una distribución lognormal, por lo que se necesita buscar nuevas distribuciones para valorar opciones emitidas sobre diferentes activos subyacentes. Varios investigadores han trabajado en nuevas fórmulas de valoración de derivados suponiendo diferentes distribuciones ya sea para el precio del activo subyacente o para su retorno. Este artículo presenta dos fórmulas para valoración de activos: una modifica la fórmula usando una distribución de Weibull de dos parámetros propuesta por Savickas (2002) añadiendo dos nuevos parámetros (escala y localización) y otra supone que la distribución del activo es una mixtura de distribuciones de Weibull. Se presentan también comparaciones de estos modelos con otros ya existentes como Black-Scholes y el modelo de Savickas con distribución Weibull simple.
Using Tukey’s g and h family of distributions to calculate value-at-risk and conditional value-at-risk José Alfredo Jiménez, Viswanathan Arunachalam Journal of Risk, 2011 Generally, when calculating value-at-risk (VaR), little importance is attached to extreme losses because they do not adequately reflect the skewness and kurtosis of the distribution. Moreover, assuming normality in VaR tends to overestimate the VaR values for upper percentiles, while it underestimates VaR for the lower percentiles of values that correspond to more extreme events. We propose to use Tukey's g and h family of distributions for calculating VaR and conditional valueat- risk (CVaR), as this distribution is able to take skewness and kurtosis into account.We also calculate an explicit formula for CVaR using the Cornish-Fisher approximation. An illustrative example is presented to compare our model with other models.
Análisis de la distribución de las tasas de retorno accionarias haciendo uso de la distribución G y H de Tukey Cuadernos De Economia, 2010
An estimation of the parameter of the g tukey distribution Revista Colombiana De Estadistica, 2006
A criterion to identify outliers Revista Colombiana De Estadistica, 2004
A generalization of the Cook's statistic Revista Colombiana De Estadistica, 2001
A proposal for the maximization of the Qk statistic Revista Colombiana De Estadistica, 2001
