Boundary–crossing probabilities for stochastic processes and their applications

In this thesis, we focus on the problem that a stochastic process crossing (or not crossing) upper and/or lower deterministic boundaries and its application in statistics, inventory management, finance, risk and ruin theory and queueing. In Chapter 2, we provide a fast and accurate method based on fast Fourier transform (FFT), to compute the (complementary) cumulative distribution function (CDF) of the Kolmogorov-Smirnov (KS) statistic when the CDF under the null hypothesis, F(x), is purely discrete, mixed or continuous, and thus obtain exact p values of the KS test. Secondly, we developed a C++ and an R implementation of the proposed method, which fills in the existing gap in statistical software. The numerical performance of the proposed FFT-based method, implemented both in C++ and in the R package KSgeneral, available from https://CRAN.R project.org/package=KSgeneral, is illustrated when F(x) is mixed, purely discrete, and continuous. In Chapter 3, we develop an efficient method based on FFT, for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries. In Chapter 4, we give explicit formulas and a numerically efficient FFT-based method for computing the probability that a non-decreasing, pure jump stochastic process will first exit from above the strip between two deterministic, possibly discontinuous, time-dependent boundaries, within a finite-time interval with an overshoot (not) exceeding a positive value. The stochastic process is a compound process with events of interest arriving according to an arbitrary point process with conditional stationary independent increments (PPCSII), and event severities with any possibly dependent joint distribution. The class of PPCSII is rather rich covering point processes with independent increments (among which non homogeneous Poisson processes and negative binomial processes), doubly stochastic Poisson (i.e., Cox processes) including mixed Poisson processes (among which processes with the order statistics property) and Markov modulated point processes. These assumptions make our framework and results generally applicable for a broad range of models arising in insurance, finance, queueing, economics, physics, astronomy and many other fields. We present examples of such applications in queueing, ruin and inventory management optimization, leading to new results in the latter fields, illustrated also numerically. In Chapter 5, we consider the large class of PPCSII and the family GD of random variables with arbitrary, possibly dependent joint distribution. These families are interchangeably used to model customers arrival and service times in the very general framework of GD/PPCSII/1 and its inverse PPCSII/GD/1 queueing models. The latter cover well known models, e.g. the G/M/1 and M/G/1 queues, but also models incorporating dependence in the arrival times, service times and across, either by directly stating their joint distribution, through a copula and appropriate marginals, or through the PPCSII class. We further introduce a double–boundary crossing (DBC) queueing duality that extends the known Cramér–Lundberg – G/M/1 duality. The DBC–queueing duality is used to establish new results with respect to the joint and marginal distributions of the busy period, idle time and the maximum waiting time, including bounds, approximations and closed form formulas. We present a FFT-based method for efficient computation of the latter distributions. We also formulate and solve novel profit optimization problems, e.g., of determining the optimal capacity of the server so as to maximize the worse-case profit margin jointly with its related probability. Results are illustrated numerically

The W,ZW,Z scale functions kit for first passage problems of spectrally negative Levy processes, and applications to the optimization of dividends

First passage problems for spectrally negative L\'evy processes with possible absorbtion or/and reflection at boundaries have been widely applied in mathematical finance, risk, queueing, and inventory/storage theory. Historically, such problems were tackled by taking Laplace transform of the associated Kolmogorov integro-differential equations involving the generator operator. In the last years there appeared an alternative approach based on the solution of two fundamental "two-sided exit" problems from an interval (TSE). A spectrally one-sided process will exit smoothly on one side on an interval, and the solution is simply expressed in terms of a "scale function" $W$ (Bertoin 1997). The non-smooth two-sided exit (or ruin) problem suggests introducing a second scale function $Z$ (Avram, Kyprianou and Pistorius 2004). Since many other problems can be reduced to TSE, researchers produced in the last years a kit of formulas expressed in terms of the "$W,Z$ alphabet" for a great variety of first passage problems. We collect here our favorite recipes from this kit, including a recent one (94) which generalizes the classic De Finetti dividend problem. One interesting use of the kit is for recognizing relationships between apparently unrelated problems -- see Lemma 3. Last but not least, it turned out recently that once the classic $W,Z$ are replaced with appropriate generalizations, the classic formulas for (absorbed/ reflected) L\'evy processes continue to hold for: a) spectrally negative Markov additive processes (Ivanovs and Palmowski 2012), b) spectrally negative L\'evy processes with Poissonian Parisian absorbtion or/and reflection (Avram, Perez and Yamazaki 2017, Avram Zhou 2017), or with Omega killing (Li and Palmowski 2017)

Building Loss Models

This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing

actuar: An R Package for Actuarial Science

actuar is a package providing additional Actuarial Science functionality to the R statistical system. The project was launched in 2005 and the package is available on the Comprehensive R Archive Network since February 2006. The current version of the package contains functions for use in the fields of loss distributions modeling, risk theory (including ruin theory), simulation of compound hierarchical models and credibility theory. This paper presents in detail but with few technical terms the most recent version of the package

Polynomial approximations for bivariate aggregate claims amount probability distributions

A numerical method to compute bivariate probability distributions from their Laplace transforms is presented. The method consists in an orthogonal projection of the probability density function with respect to a probability measure that belongs to a Natural Exponential Family with Quadratic Variance Function (NEF-QVF). A particular link to Lancaster probabilities is highlighted. The procedure allows a quick and accurate calculation of probabilities of interest and does not require strong coding skills. Numerical illustrations and comparisons with other methods are provided. This work is motivated by actuarial applications. We aim at recovering the joint distribution of two aggregate claims amounts associated with two insurance policy portfolios that are closely related, and at computing survival functions for reinsurance losses in presence of two non-proportional reinsurance treaties