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Boundary–crossing probabilities for stochastic processes and their applications

Tan, S. (2019). Boundary–crossing probabilities for stochastic processes and their applications. (Unpublished Doctoral thesis, City, University of London)

Abstract

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.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: Business School
Business School > Actuarial Science & Insurance
Doctoral Theses
Date Deposited: 29 Oct 2020 13:48
URI: https://openaccess.city.ac.uk/id/eprint/25172
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