Selected topics in financial engineering: first-exit times and dependence structures of Marshall-Olkin Kind

146 p.En esta tesis hemos investigado los tiempos de parada en diferentes ámbitos de las matemáticas financieras. Por una parte, hemos implementado una técnica de Montecarlo precisa, técnica del Puente Browniano, que estima las probabilidades de tiempos de parada de un proceso estocástico de difusión con saltos, considerando el tamaño de los saltos aleatorio y dos barreras constantes entre las cuales se mueve el proceso de difusión. Por otra parte, hemos analizado la probabilidad de distribución de la suma de los tiempos de default, dependientes entre sí, mediante la ley de probabilidad de Marshall¿Olkin. La distribución de Marshall¿Olkin es crucial en el ámbitos de la relatividad y en las aplicaciones de life-testing. Hemos derivado expresiones cerradas para la suma de los tiempos de default en el caso general bivariante y para dimensiones pequeñas considerando la familia intercambiable de la distribución de Marshall¿Olkin. Cuando la dimensión de la suma de los tiempos de default tiende a infinito, hemos demostrado que esta media converge al funcional exponencial del subordinador de Lévy. Finalmente, hemos investigado diferentes técnicas numéricas para simular las cópulas de Lévy-frailty construidas a partir de un subordinador ¿-estable de Lévy. La posibilidad de simular estas cópulas de forma precisa y rápida nos permite calcular numéricamente y de manera eficiente, el funcional exponencial del subordinador ¿-estable de Lévy

Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences

This paper deals with a surprising connection between exchangeable distributions on {0, 1}n and the recently introduced Lévy-frailty copulas, the link being provided by a new class of multivariate distribution functions called linearly order symmetric. The characterisation theorem for Lévy-frailty copulas is given a new and short (non-combinatorial) proof, and a related result is shown for exchangeable Marshall–Olkin distributions. A common thread in all these considerations is higher order monotonic functions on integer intervals of the form {0, 1, . . . , n}

Monotonicity properties of multivariate distribution and survival functions — With an application to Lévy-frailty copulas

The monotonicity properties of multivariate distribution functions are definitely more complicated than in the univariate case. We show that they fit perfectly well into the general theory of completely monotone and alternating functions on abelian semigroups. This allows us to prove a correspondence theorem which generalizes the classical version in two respects: the function in question may be defined on rather arbitrary product sets in Rn, and it need not be grounded, i.e. disappear at the lower-left boundary. In 2009 a greatly interesting class of copulas was discovered by Mai and Scherer (cf. Mai and Scherer (2009) [4]), connecting in a very surprising way complete monotonicity with respect to the maximum operation on Rn+ and with respect to ordinary addition on N0. Based on the preceding results, we give another proof of this result

Monotonicity properties of multivariate distribution and survival functions -- With an application to Lévy-frailty copulas

The monotonicity properties of multivariate distribution functions are definitely more complicated than in the univariate case. We show that they fit perfectly well into the general theory of completely monotone and alternating functions on abelian semigroups. This allows us to prove a correspondence theorem which generalizes the classical version in two respects: the function in question may be defined on rather arbitrary product sets in , and it need not be grounded, i.e. disappear at the lower-left boundary. In 2009 a greatly interesting class of copulas was discovered by Mai and Scherer (cf. Mai and Scherer (2009) [4]), connecting in a very surprising way complete monotonicity with respect to the maximum operation on and with respect to ordinary addition on . Based on the preceding results, we give another proof of this result.Multivariate distribution function Multivariate survival function Completely monotone Completely alternating n-increasing Fully n-increasing n-max-increasing Correspondence theorem Levy-frailty copula