Semimartingales, p-variational Ito-Föllmer Calculus, and Rough Path Theory
(2022) In Master's Theses in Mathematical Sciences MATM03 20222Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
- Abstract
- In this master's thesis we study stochastic integration in three parts --- through continuous semimartingales, Föllmer's deterministic $p$-variational Ito calculus, and rough path theory --- to answer the question whether stochastic integrals can be formulated as pathwise limits instead of limits in probability. The first part covers everything from discrete martingales to stochastic differential equations driven by continuous semimartingales, where we differ from the standard literature by placing a bigger emphasis on uniform integrability and equivalences thereof. We provide a slightly modified formulation of the optional stopping theorem in the uniformly integrable case, with our own proof. In the second part we review a number of... (More)
- In this master's thesis we study stochastic integration in three parts --- through continuous semimartingales, Föllmer's deterministic $p$-variational Ito calculus, and rough path theory --- to answer the question whether stochastic integrals can be formulated as pathwise limits instead of limits in probability. The first part covers everything from discrete martingales to stochastic differential equations driven by continuous semimartingales, where we differ from the standard literature by placing a bigger emphasis on uniform integrability and equivalences thereof. We provide a slightly modified formulation of the optional stopping theorem in the uniformly integrable case, with our own proof. In the second part we review a number of papers on pathwise stochastic integration, where we generalize Föllmer's deterministic approach to Ito calculus using the notion of finite $p$-variation along a sequence of partitions. In the third and final part we provide a gentle introduction to Lyons' recent theory of rough paths aimed to answer the questions posed in the second part, culminating in the result that stochastic differential equations coincide with rough differential equations, showing that stochastic integration can be formulated as a purely pathwise theory. (Less)
- Popular Abstract (Swedish)
- Ordinära differentialekvationer används ofta inom naturvetenskapliga ämnen för att modellera fysiska fenomen. Lösningarna till dessa ekvationer är deterministiska i den mening att vi alltid vet var ett system kommer hamna i framtiden givet ett startvillkor. I denna uppsats studerar vi teorin bakom så kallade stokastiska differentialekvationer (SDE), vilka uppstår genom att lägga till slumpmässigt brus till system av ordinära differentialsekvationer. En lösning till en SDE blir då en stokastisk variabel istället för ett reellt tal, vilket leder till en mer robust modell som tar hänsyn för flera olika möjligheter samtidigt. Sådana ekvationer uppstår till exempel inom volatila finansmarknader, eller när man studerar partikelrörelser inom... (More)
- Ordinära differentialekvationer används ofta inom naturvetenskapliga ämnen för att modellera fysiska fenomen. Lösningarna till dessa ekvationer är deterministiska i den mening att vi alltid vet var ett system kommer hamna i framtiden givet ett startvillkor. I denna uppsats studerar vi teorin bakom så kallade stokastiska differentialekvationer (SDE), vilka uppstår genom att lägga till slumpmässigt brus till system av ordinära differentialsekvationer. En lösning till en SDE blir då en stokastisk variabel istället för ett reellt tal, vilket leder till en mer robust modell som tar hänsyn för flera olika möjligheter samtidigt. Sådana ekvationer uppstår till exempel inom volatila finansmarknader, eller när man studerar partikelrörelser inom teoretisk fysik.
När man hittar lösningar till SDEer så använder man sig av egenskaperna av det slumpmässiga brus som man studerar i sitt system. Detta skapar ett stort filosofiskt problem när man ska tolka vad det menas med att observera ett utfall av en SDE för att sedan dra statistiska slutsatser om det underliggande systemet. Om vi exempelvis observerar en aktiekurs och tolkar kursen som en SDE, så kan man ställa sig frågan i vilken utsträckning detta enstaka utfall har för koppling till modellen i helhet som ska ta hänsyn till alla slumpmässiga utfall samtidigt. Vi svarar på denna fråga genom att visa att SDEer sammanfaller med en särskild klass av icke-differentierbara differentialekvationer som inte involverar slumpen. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9104069
- author
- Zozoulenko, Nikita LU
- supervisor
- organization
- course
- MATM03 20222
- year
- 2022
- type
- H2 - Master's Degree (Two Years)
- subject
- keywords
- stochastic analysis, probability, analysis, martingales, semimartingales, pathwise, rough path theory, stochastic differential equations
- publication/series
- Master's Theses in Mathematical Sciences
- report number
- LUNFMA-3133-2022
- ISSN
- 1404-6342
- other publication id
- 2022:E77
- language
- English
- id
- 9104069
- date added to LUP
- 2025-07-01 08:00:18
- date last changed
- 2025-07-01 08:00:18
@misc{9104069,
abstract = {{In this master's thesis we study stochastic integration in three parts --- through continuous semimartingales, Föllmer's deterministic $p$-variational Ito calculus, and rough path theory --- to answer the question whether stochastic integrals can be formulated as pathwise limits instead of limits in probability. The first part covers everything from discrete martingales to stochastic differential equations driven by continuous semimartingales, where we differ from the standard literature by placing a bigger emphasis on uniform integrability and equivalences thereof. We provide a slightly modified formulation of the optional stopping theorem in the uniformly integrable case, with our own proof. In the second part we review a number of papers on pathwise stochastic integration, where we generalize Föllmer's deterministic approach to Ito calculus using the notion of finite $p$-variation along a sequence of partitions. In the third and final part we provide a gentle introduction to Lyons' recent theory of rough paths aimed to answer the questions posed in the second part, culminating in the result that stochastic differential equations coincide with rough differential equations, showing that stochastic integration can be formulated as a purely pathwise theory.}},
author = {{Zozoulenko, Nikita}},
issn = {{1404-6342}},
language = {{eng}},
note = {{Student Paper}},
series = {{Master's Theses in Mathematical Sciences}},
title = {{Semimartingales, p-variational Ito-Föllmer Calculus, and Rough Path Theory}},
year = {{2022}},
}