Χρόνοι διακοπής με εφαρμογές στα χρηματοοικονομικά και την διαχείριση κινδύνων

Οικονομίδης, Ιάκωβος - Δαυίδ; Economides, David - Jacob

Stopping times with applications to finance and risk management

Doctoral Thesis

Author

Οικονομίδης, Ιάκωβος - Δαυίδ

Economides, David - Jacob

Date

2025-12

Abstract

Stochastic processes belong to the broader domain of probability theory concerning mathematical models for phenomena that evolve over time. For this reason, they play a central role in modeling problems across various disciplines, such as finance, risk management, physics, meteorology, biology, and sequential analysis. One main categorization of stochastic processes is based on the nature of the observation time of the stochastic phenomenon, which is typically classified as either discrete or continuous. Among the most well-known discrete-time processes is the random walk, while in continuous time the Poisson process and the Brownian motion (or Wiener process) stand out, both belonging to the wider class of Lévy processes. The aforementioned processes are at the core of this doctoral thesis, which focuses on their applications in finance and risk management. An important concept considered in the thesis that is related to the trajectory of a stochastic process the so called “stopping time”. A stopping time is a random variable that represents the first continuous or discrete time at which the process under study satisfies a certain condition. The role of stopping times is crucial in many fundamental results of stochastic processes and in particular in martingale theory (e.g. in Optional Stopping Theorem). The main applications of stopping times considered in this thesis are: (i) The time until a surplus process of an insurance company either exceeds a safety level or falls below zero, with reference to the topic of ruin theory. (ii) The time at which the market price of an asset crosses a certain threshold, triggering the exercise, or activation, or deactivation of a specific option contract. Chapter 1 provides an overview of stochastic processes in both discrete and continuous time and introduces some fundamental models needed for our study, such as Lévy processes and martingales. It then defines the notion of stopping times and reviews key related results. The chapter concludes with an exposition of several measure-changing techniques and their applications to stochastic processes. Chapter 2 presents the most established models describing surplus processes related to an insurance portfolio, such as the Sparre Andersen renewal risk model and the classical risk model perturbed by diffusion. Next, the concepts of ruin probability and the Gerber-Shiu discounted penalty function are introduced. For each model, methods of evaluating several quantities of interest are studied using appropriate measure-changing techniques. Closed-form expressions are provided in the special case where the claim sizes follow an exponential distribution. Numerical examples are included to illustrate the theoretical results discussed in this introductory chapter on ruin theory. Chapter 3 focuses on generalized Gerber-Shiu functions where the stopping time corresponds to the time of ruin. A method is proposed to extend existing generalized Gerber-Shiu functions into a new function that takes also into account the number of claims until ruin. This result is derived via a measure-changing technique and is applied to the Sparre Andersen model, allowing dependence between claim amounts and inter-claim times. For specific dependence structures, analytical expressions are provided, concerning special cases of the generalized Gerber-Shiu function. The chapter concludes with numerical illustrations of the theoretical findings. Chapter 4 addresses a more general framework where the surplus process is modeled by a renewal jump-diffusion process. The stopping time considered here is the first exit time of the surplus process from an interval of the form [0,b], introducing the notion of an upper barrier b (safety level). In this model, a new form of a generalized Gerber-Shiu-type function is defined, where the surplus is inspected only at claim arrival times. A special case of the proposed function enables the study of the joint distribution of the time, the number of claims, the surplus, and the aggregate claim amount until exit from [0,b]. For this function, closed-form expressions are derived for certain classes of distributions capable of approximating any positive distribution of claim sizes and inter-arrival times. The applicability of the theoretical results is supported by presenting numerical examples at the end of the chapter. Chapter 5 introduces fundamental concepts related to financial derivatives, offering a brief overview of the most common option types. Two particular cases are discussed in more details. The first concerns European barrier options, where pricing is derived via an appropriate measure-changing technique, with European call options arising as a special case. Closed-form formulas are provided under the geometric Brownian motion framework. The second case deals with American options, where the stopping time corresponds to the optimal exercise time chosen by the holder. Analytical results are obtained for the perpetual option using appropriate change of measure techniques in the case of geometric Brownian motion. The chapter concludes with illustrative examples. Chapter 6 is the final chapter of the thesis, which is focused on perpetual American options. In this chapter, one of the basic assumptions in the related bibliography, concerning continuous inspection of the underlying assets value, is replaced by a new, and in some situations more realistic assumption that is based on the concept of random inspection. This assumption aims to capture uncertainty in investors behavior, where actions may occur at random times. Under this framework, a randomly observed Lévy jump-diffusion process is considered. The general results are applied to the case (a) of geometric Brownian motion and (b) jump-diffusion process with exponentially distributed upward and downward jumps. The chapter concludes with numerical examples illustrating the applicability of the main results, emphasizing the role of the random inspection process, which is modeled by an exogenous Poisson process.

Department

Σχολή Χρηματοοικονομικής και Στατιστικής. Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

Number of pages

173

Language

Greek

URI

https://dione.lib.unipi.gr/xmlui/handle/unipi/18755

Collections

Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα