| Titre : | Quadratic BackwardStochasticDifferentialEquationswithJumps |
| Auteurs : | Nada Erraihane Dhahoua, Auteur ; Nabil khelfallah, Directeur de thèse |
| Editeur : | Biskra [Algérie] : Faculté des Sciences Exactes et des Sciences de la Nature et de la Vie, Université Mohamed Khider, 2024 |
| Langues: | Anglais |
| Mots-clés: | Backward stochastic differential equations ; Jump process ; Poisson random measure ; Brownian motion. |
| Résumé : |
In this master’s thesis, we study a BSDE with Jumps (BSDEJs in short), and prove the existence and uniquenes of the solutions, when the driver have quadratic growth in the Brownian companente and non-linear functional form with respect to the jump part, and the terminal condition is square integrable random variable. After proving the result of the existence and uniqueness of solutions under the strongest condition (Lipschitz condition). |
| Sommaire : |
Contents Dedication i Acknowledgmentii Notations andsymbolsiii TableofContentsvi Introduction 1 1 GeneralReminderofStochasticCalculus6 1.1 Preliminaries.................................... 6 1.1.1 PreliminariesofProbabilityTheory.................... 6 1.1.2 ConditionalExpectation.......................... 8 1.2 StochasticProcess................................. 9 1.2.1 Martingale.................................. 11 1.2.2 BrownianMotion.............................. 13 1.2.3 QuadraticVariation............................ 13 1.3 Itˆo’sStochasticCalculus.............................. 14 1.3.1 ItˆoIntegral................................. 14 1.3.2 Itˆo’sFormula................................ 15 1.4 SomeAuxiliaryResults............................... 16 2 BackwardStochasticDifferentialEquationswithJumps20 2.1 PureJumpL´evyProcesses............................. 20 2.1.1 BasicsonL´evyProcess........................... 20 2.1.2 StochasticIntegrationwithRespecttoL´evyProcesses......... 22 2.1.3 MartingaleRepresentationTheorem.................... 24 2.2 BSDEJswithGloballyLipschitzGenerators................... 25 2.2.1 NotationandDefinitions.......................... 25 2.2.2 BSDEJswithZeroGenerator....................... 27 2.2.3 BSDEJswithGeneratorIndependenttoY,ZandU........... 28 2.2.4 BSDEJsinGeneralCase.......................... 28 2.2.5 SpecialCase................................. 29 3 QuadraticBackwardStochasticDifferentialEquationswithJumps30 3.1 Introduction..................................... 30 3.2 Krylov’sEstimatesandItˆo-Krylov’sFormulaforBSDEJs............ 33 3.2.1 Krylov’sEstimatesinBSDEJs....................... 33 3.2.2 Itˆo–KrylovChangeofVariableFormulainBSDEJs........... 34 3.3 APrioriEstimates................................. 35 3.4 ExistenceandUniquenessoftheSolution..................... 38 3.5 SolvabilityofSomeQuadraticBSDEJs...................... 41 3.6 ComparisionTheorem............................... 45 Conclusion 48 |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| MM/1278 | Mémoire master | bibliothèque sciences exactes | Consultable |
