Multifractional Poisson process, multistable subordinator and related limit theorems

We introduce a multistable subordinator, which generalizes the stable subordinator to the case of time-varying stability index. This enables us to define a multifractional Poisson process. We study properties of these processes and establish the convergence of a continuous-time random walk to the multifractional Poisson process.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

Modeling and mitigation of supply-demand mismatches

In this thesis, I develop analytical models to price the value of supply chain investments under demand uncer¬tainty. This thesis includes three self-contained papers. In the first paper, we investigate the value of lead-time reduction under the risk of sudden and abnormal changes in demand forecasts. We first consider the risk of a complete and permanent loss of demand. We then provide a more general jump-diffusion model, where we add a compound Poisson process to a constant-volatility demand process to explore the impact of sudden changes in demand forecasts on the value of lead-time reduction. We use an Edgeworth series expansion to divide the lead-time cost into that arising from constant instantaneous volatility, and that arising from the risk of jumps. We show that the value of lead-time reduction increases substantially in the intensity and/or the magnitude of jumps. In the second paper, we analyze the value of quantity flexibility in the presence of supply-chain dis- intermediation problems. We use the multiplicative martingale model and the "contracts as reference points" theory to capture both positive and negative effects of quantity flexibility for the downstream level in a supply chain. We show that lead-time reduction reduces both supply-chain disintermediation problems and supply- demand mismatches. We furthermore analyze the impact of the supplier's cost structure on the profitability of quantity-flexibility contracts. When the supplier's initial investment cost is relatively low, supply-chain disin¬termediation risk becomes less important, and hence the contract becomes more profitable for the retailer. We also find that the supply-chain efficiency increases substantially with the supplier's ability to disintermediate the chain when the initial investment cost is relatively high. In the third paper, we investigate the value of dual sourcing for the products with heavy-tailed demand distributions. We apply extreme-value theory and analyze the effects of tail heaviness of demand distribution on the optimal dual-sourcing strategy. We find that the effects of tail heaviness depend on the characteristics of demand and profit parameters. When both the profit margin of the product and the cost differential between the suppliers are relatively high, it is optimal to buffer the mismatch risk by increasing both the inventory level and the responsive capacity as demand uncertainty increases. In that case, however, both the optimal inventory level and the optimal responsive capacity decrease as the tail of demand becomes heavier. When the profit margin of the product is relatively high, and the cost differential between the suppliers is relatively low, it is optimal to buffer the mismatch risk by increasing the responsive capacity and reducing the inventory level as the demand uncertainty increases. In that case, how¬ever, it is optimal to buffer with more inventory and less capacity as the tail of demand becomes heavier. We also show that the optimal responsive capacity is higher for the products with heavier tails when the fill rate is extremely high.

Valuing equity-linked death benefits in jump diffusion models

The paper is motivated by the valuation problem of guaranteed minimum death benefits in various equity-linked products. At the time of death, a benefit payment is due. It may depend not only on the price of a stock or stock fund at that time, but also on prior prices. The problem is to calculate the expected discounted value of the benefit payment. Because the distribution of the time of death can be approximated by a combination of exponential distributions, it suffices to solve the problem for an exponentially distributed time of death. The stock price process is assumed to be the exponential of a Brownian motion plus an independent compound Poisson process whose upward and downward jumps are modeled by combinations (or mixtures) of exponential distributions. Results for exponential stopping of a Lévy process are used to derive a series of closed-form formulas for call, put, lookback, and barrier options, dynamic fund protection, and dynamic withdrawal benefit with guarantee. We also discuss how barrier options can be used to model lapses and surrenders.

Étude empirique de distributions associées à la Fonction de Pénalité Escomptée

On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature.

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t,x→∞t,x→∞, where t>0t>0 is the time horizon and x>0x>0 is the starting point of the risk process, with y=t/xy=t/x held constant and assumed either below or above a certain constant.

Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues

In the Hammersley-Aldous-Diaconis process, infinitely many particles sit in R and at most one particle is allowed at each position. A particle at x, whose nearest neighbor to the right is at y, jumps at rate y - x to a position uniformly distributed in the interval (x, y). The basic coupling between trajectories with different initial configuration induces a process with different classes of particles. We show that the invariant measures for the two-class process can be obtained as follows. First, a stationary M/M/1 queue is constructed as a function of two homogeneous Poisson processes, the arrivals with rate, and the (attempted) services with rate rho > lambda Then put first class particles at the instants of departures (effective services) and second class particles at the instants of unused services. The procedure is generalized for the n-class case by using n - 1 queues in tandem with n - 1 priority types of customers. A multi-line process is introduced; it consists of a coupling (different from Liggett's basic coupling), having as invariant measure the product of Poisson processes. The definition of the multi-line process involves the dual points of the space-time Poisson process used in the graphical construction of the reversed process. The coupled process is a transformation of the multi-line process and its invariant measure is the transformation described above of the product measure.

POISSON DISTRIBUTION TO ANALYZE NEAR-THRESHOLD MOTOR EVOKED POTENTIALS

Motor unit action potentials (MUAPs) evoked by repetitive, low-intensity transcranial magnetic stimulation can be modeled as a Poisson process. A mathematical consequence of such a model is that the ratio of the variance to the mean of the amplitudes of motor evoked potentials (MEPs) should provide an estimate of the mean size of the individual MUAPs that summate to generate each MEP. We found that this is, in fact, the case. Our finding thus supports the use of the Poisson distribution to model MEP generation and indicates that this model enables characterization of the motor unit population that contributes to near-threshold MEPs. Muscle Nerve 42: 825-828, 2010

A direct approach to the discounted penalty function

This paper provides a new and accessible approach to establishing certain results concerning the discounted penalty function. The direct approach consists of two steps. In the first step, closed-form expressions are obtained in the special case in which the claim amount distribution is a combination of exponential distributions. A rational function is useful in this context. For the second step, one observes that the family of combinations of exponential distributions is dense. Hence, it suffices to reformulate the results of the first step to obtain general results. The surplus process has downward and upward jumps, modeled by two independent compound Poisson processes. If the distribution of the upward jumps is exponential, a series of new results can be obtained with ease. Subsequently, certain results of Gerber and Shiu [H. U. Gerber and E. S. W. Shiu, North American Actuarial Journal 2(1): 48–78 (1998)] can be reproduced. The two-step approach is also applied when an independent Wiener process is added to the surplus process. Certain results are related to Zhang et al. [Z. Zhang, H. Yang, and S. Li, Journal of Computational and Applied Mathematics 233: 1773–1 784 (2010)], which uses different methods.

Irreversible thermodynamics of Poisson processes with reaction

A kinetic model is derived to study the successive movements of particles, described by a Poisson process, as well as their generation. The irreversible thermodynamics of this system is also studied from the kinetic model. This makes it possible to evaluate the differences between thermodynamical quantities computed exactly and up to second-order. Such differences determine the range of validity of the second-order approximation to extended irreversible thermodynamics

Aplicación de colas de Poisson en procesos de ‘toma de decisiones’ en la gestión de servicios médicos

En este documento se revisa teóricamente la distribución de probabilidad de Poisson como función que asigna a cada suceso definido, sobre una variable aleatoria discreta, la probabilidad de ocurrencia en un intervalo de tiempo o región del espacio disjunto. Adicionalmente se revisa la distribución exponencial negativa empleada para modelar el intervalo de tiempo entre eventos consecutivos de Poisson que ocurren de manera independiente; es decir, en los cuales la probabilidad de ocurrencia de los eventos sucedidos en un intervalo de tiempo no depende de los ocurridos en otros intervalos de tiempo, por esta razón se afirma que es una distribución que no tiene memoria. El proceso de Poisson relaciona la función de Poisson, que representa un conjunto de eventos independientes sucedidos en un intervalo de tiempo o región del espacio con los tiempos dados entre la ocurrencia de los eventos según la distribución exponencial negativa. Los anteriores conceptos se usan en la teoría de colas, rama de la investigación de operaciones que describe y brinda soluciones a situaciones en las que un conjunto de individuos o elementos forman colas en espera de que se les preste un servicio, por lo cual se presentan ejemplos de aplicación en el ámbito médico.

Using non-homogeneous Poisson models with multiple change-points to estimate the number of ozone exceedances in Mexico City

In this paper, we consider some non-homogeneous Poisson models to estimate the probability that an air quality standard is exceeded a given number of times in a time interval of interest. We assume that the number of exceedances occurs according to a non-homogeneous Poisson process (NHPP). This Poisson process has rate function lambda(t), t >= 0, which depends on some parameters that must be estimated. We take into account two cases of rate functions: the Weibull and the Goel-Okumoto. We consider models with and without change-points. When the presence of change-points is assumed, we may have the presence of either one, two or three change-points, depending of the data set. The parameters of the rate functions are estimated using a Gibbs sampling algorithm. Results are applied to ozone data provided by the Mexico City monitoring network. In a first instance, we assume that there are no change-points present. Depending on the adjustment of the model, we assume the presence of either one, two or three change-points. Copyright (C) 2009 John Wiley & Sons, Ltd.

Studies in non-Gaussian analysis

This thesis presents general methods in non-Gaussian analysis in infinite dimensional spaces. As main applications we study Poisson and compound Poisson spaces. Given a probability measure μ on a co-nuclear space, we develop an abstract theory based on the generalized Appell systems which are bi-orthogonal. We study its properties as well as the generated Gelfand triples. As an example we consider the important case of Poisson measures. The product and Wick calculus are developed on this context. We provide formulas for the change of the generalized Appell system under a transformation of the measure. The L² structure for the Poisson measure, compound Poisson and Gamma measures are elaborated. We exhibit the chaos decomposition using the Fock isomorphism. We obtain the representation of the creation, annihilation operators. We construct two types of differential geometry on the configuration space over a differentiable manifold. These two geometries are related through the Dirichlet forms for Poisson measures as well as for its perturbations. Finally, we construct the internal geometry on the compound configurations space. In particular, the intrinsic gradient, the divergence and the Laplace-Beltrami operator. As a result, we may define the Dirichlet forms which are associated to a diffusion process. Consequently, we obtain the representation of the Lie algebra of vector fields with compact support. All these results extends directly for the marked Poisson spaces.

A non-homogeneous poisson model with spatial anisotropy applied to ozone data from Mexico City

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)