The goal of this thesis - in the context of uncertainty quantification - is to propose a numerical scheme, alternative to classical Monte Carlo methods, that allows to sample quickly from a special class of random variables keeping a high level of accuracy. Specifically we place our attention on the so-called time-integrated bridges, i.e., random variables defined as the integral over time of stochastic processes conditional on their initial and final values. The proposed scheme is entirely novel and goes into an area almost totally unexplored in the literature. The methodology combines techniques of stochastic collocation with advanced tools of deep learning to achieve a flexible scheme that opens a window into many possible research directions. The thesis provides all the details concerning the proposed method, trying to introduce them with a simple and intuitive language, but at the same time without neglecting the mathematical and scientific rigor that is expected in such a context. The first two chapters are introductory. First, in Chapter 1, general reasons are given to support the use of non-trivial mathematical tools, such as stochastic processes or their transformations. In addition, a brief review of the various results inherent in the topic and available in the literature is presented. The focus is placed on the two main sources that inspired the work: the Stocastic Collocation Monte Carlo technique and the Seven-League scheme. Chapter 2 is devoted to the description of the general model. This framework can be easily connected to two important applications from the financial environment, presented in the same chapter and then employed as examples when the procedure is actually applied. Chapters three to five are undoubtedly the heart of the project, and this is particularly true of chapter four in which the proposed innovative idea is encapsulated. In these chapters the scheme is investigated in detail and each part is explained with detail. In Chapter 3 the technique of stochastic collocation, particularly Stochastic Collocation Monte Carlo sampler, is presented. It will provide an indispensable tool to “compress” the information required for sampling from the time-integrated bridge, without deteriorating the quality of the simulation. In Chapter 4 the proposed methodology is described in its entirety: the idea of combining the Stochastic Collocation Monte Carlo technique with the use of artificial neural networks is very powerful and innovative. The two phases – “offline” and “online” - are presented highlighting the various tools used and the motivations supporting them. The scheme is easily implementable by following the explanations in the chapter, with the addition of useful bullet point algorithms available in Appendix A. In addition, in Chapter 5, a discussion is provided regarding the types of error introduced by the various approximations used, which are necessary at the implementation level. For some errors the discussion is extremely rigorous, while for others we only provide a more intuitive and, somehow, empirical discussion of the issue. Chapter 6 shows the scheme “at work”: the proposed methodology turns out to be not only extremely original, but also very efficient. The results are excellent, and particularly the show the generality of the proposed scheme. In fact, the methodology is tested on several processes, from the simple arithmetic Brownian motion to the Cox-Ingersoll-Ross process, passing through the well known geometric Brownian motion. Moreover, the experiments performed on the above models are easily declined and employed in the applications coming from the financial context introduced in Chapter 2. In all the applications presented the results are largely satisfactory, far exceeding the performance of classical Monte Carlo schemes. In particular, the innovative method proposed produces results with a considerably reduced computational time (up to more than ten times less!), but without negatively affecting the accuracy of the simulation. Finally, in Chapter 7 the project described in the thesis is summarized and some concluding remarks are given.
L'obiettivo di questa tesi - nell'ambito della quantificazione dell'incertezza - è proporre uno schema numerico, alternativo ai classici metodi Monte Carlo, che permetta di campionare da una speciale classe di variabili aleatorie velocemente e con grande precisione. Nello specifico poniamo la nostra attenzione sui cosiddetti bridge tempo-integrati, ossia variabili aleatorie definite come l'integrale nel tempo di processi stocastici condizionati alle loro condizioni iniziali e finali. Lo schema proposto è del tutto innovativo e va a collocarsi in un'area quasi totalmente inesplorata in letteratura. La metodologia combina tecniche di stochastic collocation con avanzati strumenti di deep learning per ottenere uno schema flessibile che apre uno spiraglio su molte possibili direzioni di ricerca. La tesi fornisce tutti i dettagli riguardanti il metodo proposto, cercando di introdurli con un linguaggio semplice ed intuitivo, ma al contempo senza trascurare il rigore matematico e scientifico che ci si aspetta in tale contesto. I primi due capitoli sono introduttivi. Innanzitutto, nel Capitolo 1 vengono fornite motivazioni di carattere generale a supporto dell'uso di strumenti matematici non banali, quali processi stocastici o trasformazioni degli stessi. Inoltre, viene presentata una breve rassegna dei vari risultati inerenti l'argomento e disponibili in letteratura. Viene posto il focus sulle due principali fonti che hanno ispirato il lavoro: la tecnica Stocastic Collocation Monte Carlo e il Seven-League scheme. Il Capitolo 2 è dedicato alla descrizione del modello generale. Esso viene facilmente connesso a due importanti applicazioni provenienti dall'ambiente finanziario, presentate nello stesso capitolo e poi riprese a titolo di esempio quando la procedura viene effettivamente applicata. I capitoli dal terzo al quinto sono indubbiamente il cuore del progetto, e ciò è particolarmente vero per il quarto in cui è racchiusa l'idea innovativa proposta. In questi capitoli lo schema viene indagato dettagliatamente ed ogni parte viene spiegata con perizia di particolari. Nel Capitolo 3 viene presentata la tecnica di stochastic collocation, chiamata in particolare Stochastic Collocation Monte Carlo. Essa fornirà uno strumento indispensabile per “comprimere” l'informazione necessaria per il campionamento dal bridge tempo-integrato, senza deteriorare la qualità della simulazione. Nel Capitolo 4 viene descritta la metodologia proposta nella sua totalità: l'idea di combinare la tecnica Stochastic Collocation Monte Carlo con l'uso di reti neurali artificiali è molto valida, nonché innovativa. Le due fasi – “offline” ed “online” - sono presentate mettendo in evidenza i vari strumenti utilizzati e le motivazioni a supporto di questi. Lo schema è facilmente implementabile seguendo le spiegazioni riportate nel capitolo, con l'aggiunta di utili algoritmi puntuali disponibili in Appendice A. Inoltre, nel Capitolo 5, viene fornita una discussione riguardo i tipi di errore introdotti dalle varie approssimazioni impiegate, necessarie a livello implementativo. Per alcuni errori la trattazione è estremamente rigorosa, mentre per altri ci limitiamo ad una discussione più intuitiva e, se vogliamo, empirica della questione. Il Capitolo 6 mostra lo schema “all'opera”: la metodologia proposta si rivela, oltre che estremamente originale, anche molto efficiente. I risultati sono eccellenti, e soprattutto mostrano il carattere generale dello schema proposto. Esso viene infatti testato su diversi processi integrandi, dal semplice moto Browniano aritmetico al processo di Cox-Ingersoll-Ross, passando per il ben noto moto Browniano geometrico. Inoltre, gli esperimenti eseguiti sui modelli sopracitati sono facilmente declinati ed impiegati nelle applicazioni provenienti dal contesto finanziario introdotte nel Capitolo 2. In tutte le applicazioni i risultati sono ampiamente soddisfacenti, superando di gran lunga le prestazioni dei classici schemi Monte Carlo. In particolare, l'innovativo metodo proposto produce risultati con un tempo di calcolo considerevolmente ridotto (addirittura fino a più di dieci volte inferiore!), senza però influenzare negativamente l'accuratezza della simulazione. Infine, nel Capitolo 7 viene riassunto il progetto descritto nella tesi e vengono riportate alcune considerazioni di carattere conclusivo.
Fast sampling from time-integrated bridges using deep learning
PEROTTI, LEONARDO
2020/2021
Abstract
The goal of this thesis - in the context of uncertainty quantification - is to propose a numerical scheme, alternative to classical Monte Carlo methods, that allows to sample quickly from a special class of random variables keeping a high level of accuracy. Specifically we place our attention on the so-called time-integrated bridges, i.e., random variables defined as the integral over time of stochastic processes conditional on their initial and final values. The proposed scheme is entirely novel and goes into an area almost totally unexplored in the literature. The methodology combines techniques of stochastic collocation with advanced tools of deep learning to achieve a flexible scheme that opens a window into many possible research directions. The thesis provides all the details concerning the proposed method, trying to introduce them with a simple and intuitive language, but at the same time without neglecting the mathematical and scientific rigor that is expected in such a context. The first two chapters are introductory. First, in Chapter 1, general reasons are given to support the use of non-trivial mathematical tools, such as stochastic processes or their transformations. In addition, a brief review of the various results inherent in the topic and available in the literature is presented. The focus is placed on the two main sources that inspired the work: the Stocastic Collocation Monte Carlo technique and the Seven-League scheme. Chapter 2 is devoted to the description of the general model. This framework can be easily connected to two important applications from the financial environment, presented in the same chapter and then employed as examples when the procedure is actually applied. Chapters three to five are undoubtedly the heart of the project, and this is particularly true of chapter four in which the proposed innovative idea is encapsulated. In these chapters the scheme is investigated in detail and each part is explained with detail. In Chapter 3 the technique of stochastic collocation, particularly Stochastic Collocation Monte Carlo sampler, is presented. It will provide an indispensable tool to “compress” the information required for sampling from the time-integrated bridge, without deteriorating the quality of the simulation. In Chapter 4 the proposed methodology is described in its entirety: the idea of combining the Stochastic Collocation Monte Carlo technique with the use of artificial neural networks is very powerful and innovative. The two phases – “offline” and “online” - are presented highlighting the various tools used and the motivations supporting them. The scheme is easily implementable by following the explanations in the chapter, with the addition of useful bullet point algorithms available in Appendix A. In addition, in Chapter 5, a discussion is provided regarding the types of error introduced by the various approximations used, which are necessary at the implementation level. For some errors the discussion is extremely rigorous, while for others we only provide a more intuitive and, somehow, empirical discussion of the issue. Chapter 6 shows the scheme “at work”: the proposed methodology turns out to be not only extremely original, but also very efficient. The results are excellent, and particularly the show the generality of the proposed scheme. In fact, the methodology is tested on several processes, from the simple arithmetic Brownian motion to the Cox-Ingersoll-Ross process, passing through the well known geometric Brownian motion. Moreover, the experiments performed on the above models are easily declined and employed in the applications coming from the financial context introduced in Chapter 2. In all the applications presented the results are largely satisfactory, far exceeding the performance of classical Monte Carlo schemes. In particular, the innovative method proposed produces results with a considerably reduced computational time (up to more than ten times less!), but without negatively affecting the accuracy of the simulation. Finally, in Chapter 7 the project described in the thesis is summarized and some concluding remarks are given.| File | Dimensione | Formato | |
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Descrizione: Tesi di laurea magistrale di Leonardo Perotti
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https://hdl.handle.net/10589/177616