[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121712-en":3,"doc-seo-121712-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121712,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Quantitative Finance informed Machine Learning - Doctor of Philosophy","This PhD thesis addresses quantitative finance problems using deep learning to approximate high-dimensional path-dependent parabolic linear PDEs and to learn model dynamics from market data with strong functional priors. It proposes deep solvers that approximate both PDE solutions and gradients, including a bias-removal approach via gradient-based network estimates combined with Monte Carlo. A second part develops Neural SDEs and Sig-Wasserstein GANs for time series, supported by extensive numerical experiments demonstrating algorithmic efficiency.","This thesis has been submitted in fulfilment of the requirements for a postgraduate degree (e. g. PhD, MPhil, DClinPsychol) at the University of Edinburgh. Please note the following terms and conditions of use:  \n• This work is protected by copyright and other intellectual property rights, which are retained by the thesis author, unless otherwise stated.  \n• A copy can be downloaded for personal non-commercial research or study, without prior permission or charge.  \n• This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the author.  \n• The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the author.  \n• When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given.  \nQuantitative Finance informed Machine  \nLearning  \nMarc Sabate´ Vidales  \nDoctor of Philosophy University of Edinburgh  \nDeclaration  \nI declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text.  \n(Marc Sabate´ Vidales)  \n4  \nAbstract  \nThis PhD thesis consists of two parts. In the first part, we develop and study deep learning-based methods for approximating high-dimensional parabolic (path-dependent) linear PDEs parametrised by the model parameters. The proposed algorithms approximate solutions together with their gradients. Furthermore, we show how to remove the bias in the deep network approximation of the PDE solution by combining obtained approximation of the gradient of the PDE with Monte Carlo simulations. In the context of quantitative finance, our methodology can be used to simultaneously compute the prices and hedging strategies for (path-dependent) derivatives with the underlying modelled by a diffusion model with possible path-dependent coefficients across ranges of parameters and initial conditions.  \nIn the second part of this thesis, we study techniques that enable learning the model using market data while keeping a strong prior on its form. First, we develop Neural SDEs, which is a stochastic differential equation (SDE) where the drift and the diffusion coefficients are parametrised by Neural Networks. This allows us to obtain robust bounds for prices of derivatives while incorporating relevant market data. Neural SDEs can also be seen as an instance of generative models, a class of Machine Learning models that learn to sample from a target distribution (either time marginal or on the path space) . Second, we shift our focus to conditional generators for time series data. We introduce the Sig-Wasserstein GAN that relies on a path signature that emerged from rough path theory and enjoyed universal approximation property. All the results of the thesis are underpinned with thorough numerical experiments that show the efficiency of our algorithms.  \n6  \nContents  \nAbstract 5  \n1 Introduction 13  \n1.1 Machine learning methods applied to Mathematical Finance-Option pricing ..... 13  \n1.2 Thesis outline ....................................... 15  \n1.2.1 High-dimensional PDE approximation ...................... 15  \n1.2.2 Time series generative methods-Data-driven market generator ......... 16  \n1.3 Code ............................................ 16  \n2 Unbiased deep solvers for linear parametric deep PDEs 17  \n2.1 Introduction ........................................ 17  \n2.1.1 Main contributions ................................ 18  \n2.1.2 Literature review ................................. 19  \n2.1.3 Notation ..................................... 20  \n2.1.4 Outline ...................................... 20  \n2.2 PDE Martingale control variate .............................. 21  \n2.2.1 PDE derivation of the control variate ....................... 22  \n2.2.2 Unbiased Parametric PDE approximation .................... 23  \n2.3 Deep PDE solvers ...................","cbCaipO9qQ9ABuoE","https://ap.wps.com/l/cbCaipO9qQ9ABuoE","pdf",3300665,1,123,"English","en",105,"# Abstract\n# 1 Introduction\n## 1.1 Machine learning methods applied to Mathematical Finance-Option pricing\n## 1.2 Thesis outline\n## 1.3 Code\n# 2 Unbiased deep solvers for linear parametric deep PDEs\n## 2.1 Introduction\n## 2.2 PDE Martingale control variate\n## 2.3 Deep PDE solvers\n## 2.4 Examples and experiments\n# 3 Solving path dependent PDEs with LSTM networks and path signatures","[{\"question\":\"What is the main focus of the first part of the thesis?\",\"answer\":\"It develops and studies deep learning-based methods to approximate high-dimensional path-dependent parabolic linear PDEs, including approximations of solutions and their gradients.\"},{\"question\":\"How does the thesis remove bias in deep network approximations?\",\"answer\":\"It combines the learned gradient approximation with Monte Carlo simulations to remove bias in the PDE solution approximation.\"},{\"question\":\"What methods are introduced in the second part for learning from market data?\",\"answer\":\"It introduces Neural SDEs to learn drift and diffusion through neural networks, and Sig-Wasserstein GANs for time series using path signatures from rough path theory.\"}]","Quantitative Finance informed Machine Learning - Doctor of Philosophy | PDF",1785806432,310,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"quantitative-finance-informed-machine-learning-doctor-of-philosophy","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/quantitative-finance-informed-machine-learning-doctor-of-philosophy/121712/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main focus of the first part of the thesis?","Question",{"text":75,"@type":76},"It develops and studies deep learning-based methods to approximate high-dimensional path-dependent parabolic linear PDEs, including approximations of solutions and their gradients.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the thesis remove bias in deep network approximations?",{"text":80,"@type":76},"It combines the learned gradient approximation with Monte Carlo simulations to remove bias in the PDE solution approximation.",{"name":82,"@type":73,"acceptedAnswer":83},"What methods are introduced in the second part for learning from market data?",{"text":84,"@type":76},"It introduces Neural SDEs to learn drift and diffusion through neural networks, and Sig-Wasserstein GANs for time series using path signatures from rough path theory.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]