Discrete copulas and quasi-copulas : graduation seminar work

Ivanova, Lina (author), Stopar, Nik (mentor)

The thesis gives a brief introduction to two dimensional copulas and related functions. Firstly, we present copulas, their usage and according history. Furthermore, in the text we include some examples and visual representations for this purpose. Copulas are cumulative distribution functions defined on the unit square with marginal distributions which are uniformly distributed on the unit interval. The motivation for this topic comes from the fact that this functions are used for describing dependency structures and they represent better tool for this because of the non-linear nature of the variables. Historically, copulas were introduced in the late 50s and quasi-copulas in the 90s which makes them relatively new concept in mathematics with high potential for further analysis. Moreover, after the 90s they got quite popular in the finance world and still today are used in risk management and while modelling returns. An important property of copulas is that they are closely connected to measure theory. Namely, every copula induces a bistochastic measure on the Borel sigma algebra of the unit square and vice versa every bistochastic measure corresponds to some copula. Hence, these functions are measuring something and can be seen as an assignment of a number between 0 and 1 to every rectangle in the unit square. If one discretises the domain of such a function then one can get the so called discrete copulas, which are functions that satisfies similar properties as copulas. They are often useful because it is easier to operate with them. It is possible to construct a copula with discrete copula by extending it from discrete domain to a continuous via piecewise bilinear interpolation. In the last part of the thesis we consider quasi-copulas which are more general functions than copulas. Accordingly, they are a functions defined on a same domain as copulas and have some (but not all) of the properties of copulas. Similarly, one can discretise a quasi-copula in order to get a discrete quasi-copula.

• Language: English
• Keywords: copula, quasi-copula, bistochastic measure, bistochastic matrix
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2022
• PID: 20.500.12556/RUL-140494
• UDC: 519.2
• COBISS.SI-ID: 126554115
• Publication date in RUL: 15.09.2022

Secondary language

• Language: Slovenian
• Title: Diskretne kopule in kvazi-kopule

Abstract

Diplomsko delo na kratko predstavi dvodimenzionalne kopule in sorodne funkcije. Najprej predstavimo kopule, njihovo uporabo in zgodovino. Poleg tega smo v ta namen v seminar vključili nekaj primerov in vizualnih prikazov. Kopule so kumulativne porazdelitvene funkcije, definirane na enotskem kvadratu z marginalnimi porazdelitvami, ki so enakomerno porazdeljene na enotskem intervalu. Motivacija za to temo izhaja iz dejstva, da se te funkcije uporabljajo za opisovanje struktur odvisnosti in so zaradi nelinearne narave spremenljivk boljše orodje od Pearsonovega korelacijskega koeficienta za to. Zgodovinsko gledano so bile kopule vpeljane v poznih 50. letih, kvazi-kopule pa v 90. letih, kar pomeni, da predstavljajo relativno nov koncept v matematiki z velikim potencialom za nadaljnjo analizo. Poleg tega so po 90. letih postale precej priljubljene v finančnem svetu in se še danes uporabljajo pri upravljanju tveganj in modeliranju donosov. Pomembna lastnost kopul je, da so tesno povezane s teorijo mere. Namreč, vsaka kopula inducira bistohastično mero na Borelovi sigma algebri enotskega kvadrata in obratno, vsaka bistohastična mera ustreza neki kopuli. Te funkcije torej nekaj merijo in jih lahko razumemo kot dodelitev števila med 0 in 1 vsakemu pravokotniku v enotskem kvadratu. Če domeno take funkcije diskretiziramo, lahko dobimo tako imenovane diskretne kopule, ki so funkcije s podobnimi lastnostmi kot kopule. Diskretne kopule se pogosto uporabljajo zato, ker je z njimi lažje delati. Običajno kopulo lahko iz diskretne kopule konstruiramo tako, da jo z diskretne domene razširimo na cel kvadrat s pomočjo kosoma bilinearne interpolacije. V zadnjem delu seminarja obravnavamo kvazi-kopule, ki so splošnejše funkcije od kopul. Kvazi-kopule so funkcije definirane na isti domeni kot kopule in imajo nekatere (vendar ne vseh) lastnosti kopul. Tudi kvazi-kopule lahko diskretiziramo, da dobimo diskretne kvazi-kopule.

• Keywords: kopula, kvazi-kopula, bistohastična mera, bistohastična matrika