Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations

Detalhes bibliográficos
Autor(a) principal: Gomes, André de Oliveira
Data de Publicação: 2011
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10451/8447
Resumo: Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2011
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spelling Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equationsExistence and uniquenessForward backward stochastic differentialGradient estimatesQuasilinear equations of parabolic typeFour step schemeLarge deviationsFreidlin-Wentzell theoryBurgers equations typeViscosity solutionsTeses de mestrado - 2011Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2011We consider Forward Backward Stochastic Differential Equations (FBSDEs for short) with different assumptions on its coefficients. In a first part we present results of existence, uniqueness and dependence upon initial conditions and on the coefficients. There are two main methodologies employed in this study. The first one presented is the Four Step Scheme, which makes very clear the connection of FBSDEs with quasilinear parabolic systems of Partial Differential Equations (PDEs for short). The weakness of this methodology is the smoothness and regularity assumptions recquired on the coefficients of the system, which motivate the employment of Banach`s Fixed Point Theorem in the study of existence and uniqueness results. This classic analytical tool requires less regularity on the coefficients, but gives only local existence of solution in a small time duration. In a second stage, with the help of the previous work with a running-down induction on time, we can assure the existence and uniqueness of solution for the FBSDE problem in global time. The second goal of this work is the study of the assymptotic behaviour of the FBSDEs solutions when the diffusion coefficient of the forward equation is multiplicatively perturbed with a small parameter that goes to zero. This question adresses the problem of the convergence of the classical/viscosity solutions of the quasilinear parabolic system of PDEs associated to the system. When this quasilinear parabolic system of PDEs takes the form of the backward Burgers Equation, the problem is the convergence of the solution when the viscosity parameter goes to zero. To study conveniently this problem with a probabilistic approach , we present a concise survey of the classical Large Deviations Principles and the basics of the so-called "Freidlin-Wentzell Theory". This theory is mainly concerned with the study of the Itô Diffusions with the diffusion term perturbed by a small parameter that converges to zero and the richness of properties of the FBSDEs shows us that (even in a coupled FBSDE system) this approach is a good one, since we can extract for the solutions of the perturbed systems a Large Deviations Principle and state convergence of the perturbed solutions to a solution of a deterministic system of ordinary differential equations.Cruzeiro, Ana Bela, 1957-Zambrini, Jean Claude, 1951-Repositório da Universidade de LisboaGomes, André de Oliveira2013-05-13T14:46:36Z20112011-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttp://hdl.handle.net/10451/8447enginfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-08T15:52:20Zoai:repositorio.ul.pt:10451/8447Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:32:59.543617Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
title Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
spellingShingle Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
Gomes, André de Oliveira
Existence and uniqueness
Forward backward stochastic differential
Gradient estimates
Quasilinear equations of parabolic type
Four step scheme
Large deviations
Freidlin-Wentzell theory
Burgers equations type
Viscosity solutions
Teses de mestrado - 2011
title_short Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
title_full Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
title_fullStr Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
title_full_unstemmed Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
title_sort Forward backward stochastic differential equations: existence, uniqueness, a large deviations principle and connections with partial differential equations
author Gomes, André de Oliveira
author_facet Gomes, André de Oliveira
author_role author
dc.contributor.none.fl_str_mv Cruzeiro, Ana Bela, 1957-
Zambrini, Jean Claude, 1951-
Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Gomes, André de Oliveira
dc.subject.por.fl_str_mv Existence and uniqueness
Forward backward stochastic differential
Gradient estimates
Quasilinear equations of parabolic type
Four step scheme
Large deviations
Freidlin-Wentzell theory
Burgers equations type
Viscosity solutions
Teses de mestrado - 2011
topic Existence and uniqueness
Forward backward stochastic differential
Gradient estimates
Quasilinear equations of parabolic type
Four step scheme
Large deviations
Freidlin-Wentzell theory
Burgers equations type
Viscosity solutions
Teses de mestrado - 2011
description Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2011
publishDate 2011
dc.date.none.fl_str_mv 2011
2011-01-01T00:00:00Z
2013-05-13T14:46:36Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10451/8447
url http://hdl.handle.net/10451/8447
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
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