Expected Discounted Penalty Functions and Wiener-Hopf factorization for two-sided jumps Lévy risk processes

• Autores: Ehyter Matías Martín González
• Directores de la Tesis: Ekaterina Todorova Kolkovska (dir. tes.)
• Lectura: En la Centro de Investigación en Matemáticas (CIMAT) ( México ) en 2016
• Idioma: inglés
• Enlaces
  • Tesis en acceso abierto en: cimat.repositorioinstitucional.mx
• Resumen
  • español

    Consideramos la clase de procesos de Levy X = fX(t); t 0g denidos mediante expresiones de la forma X(t) = ct + B(t) + Z(t) S(t);

    0;

    donde c 0 es un termino de deriva, B = fB(t); t 0g es un movimiento browniano con media cero, Z = fZ(t); t 0g es un proceso Poisson compuesto tal que la magnitud de sus saltos tienen transformada de Laplace racional y S = fS(t); t 0g es un proceso de Levy de puros saltos, con saltos positivos. Suponemos que B, Z y S son independientes.

    En este trabajo estudiamos la factorizacion de Wiener-Hopf de esta clase de procesos. En particular nos enfocamos en el factor negativo de Wiener-Hopf, que corresponde al nmo del proceso tomado en el intervalo de cero hasta un tiempo aleatorio exponencial.

    Identicamos explcitamente el subordinador asociado a este factor negativo, y utilizamos este resultado para obtener una expresion para la funcion de densidad de dicho factor. Esta densidad esta expresada en terminos de los parametros del proceso y funciones conocidas; por ejemplo, una funcion q-escala de un proceso de Levy espectralmente negativo asociado a X.

    Utilizamos los resultados anteriores para obtener una expresion para la Funcion de penalidad descontada esperada generalizada (o funcion de Gerber-Shiu generalizada) asociada a procesos de la forma u + X, donde u 0.

    Estudiamos tambien un caso particular del proceso u+X, que corresponde a un proceso de riesgo con saltos positivos y negativos, perturbado por un movimiento -estable. Para este caso particular se obtuvieron expresiones asintoticas para la probabilidad de ruina y para la cola conjunta de la severidad de la ruina y el capital previo a la ruina.

  • English

    Let us consider the class of L´evy processes X = {X (t), t ≥ 0} defined by the equation X (t) = ct + γB(t) + Z(t) − S(t), γ ≥ 0, where c ≥ 0 is a drift term, B = {B(t), t ≥ 0} is a brownian motion with zero mean, Z = {Z(t), t ≥ 0} is a compound Poisson process whose jumps have a probability distribution with rational Laplace transform and S = {S(t), t ≥ 0} is a pure positive jumps L´evy process.

    The processes B, Z and S are assumed to be independent.

    We study the Wiener-Hopf factorization of this class of processes, particularly focusing on the distribution of the negative Wiener-Hopf factor (the factor given by the infimum of X stopped at a random exponential time). We present explicitly a subordinator such that the negative Wiener-Hopf factor is equal in distribution to this subordinator, and then use this result to obtain an expression for the Laplace transform of this negative Wiener-Hopf factor.

    We invert this Laplace transform to obtain an explicit expression for its probability density.

    This probability density is in terms of functions which depend only on the parameters of the process, and in terms of the derivative of a q-scale function of an associated spectrally negative L´evy process. Due to the importance of q-scale functions, we apply our techniques to study two important cases of spectrally negative L´evy processes and obtain explicit expressions for their corresponding q-scale functions.

    Furthermore, we use our result about the density of the negative Wiener-Hopf factor to obtain an explicit expression for the Generalized Expected Discounted Penalty Function (Generalized EDPF, for short) of a process of the form u + X , where u ≥ 0. This model corresponds to a two-sided jumps L´evy risk process with rational positive jumps and general negative jumps, allowing the case when there exists a random factor (a perturbation) which models random gains or losses. The two-sided jumps L´evy risk process with an α-stable perturbation, which generalizes the model in Furrer [1998], arises as a particular case. This model is studied in full detail and we present several results about its corresponding EDPF.

    We also obtain asymptotic expressions for its corresponding ruin probability and the joint tail of the severity of ruin and the surplus prior to ruin for this case. The asymptotic results for this joint tail are also stated for the classical risk process perturbed by an αstable motion.