Stochastic differential equations : strong well-posedness of singular and degenerate equations; numerical analysis of decoupled forward backward systems of McKean-Vlasov type
type: Dataset
description: This thesis deals with two subjects: the strong well-posedness of stochastic differential equations with Hölder drift and hypoelliptic noise and the simulation of decoupled forward backward stochastic differential equations of McKean-Vlasov type. In the first work, we study a class of degenerate system with hypoelliptic noise. We prove that strong well-posedness holds for this system when the drift is only H\"{o}lder, with Hölder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin (1974), Veretennikov (1980) and Krylov and Röckner (2005). The existence of a threshold for the Hölder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method. In the second work, we propose a new algorithm to approach weakly the solution of a McKean-Vlasov stochastic differential equation. Based on the cubature method, the algorithm is deterministic differing from the usual methods based on interacting particles. It can be parametrized in order to obtain a given order of convergence. Then, we construct implementable algorithms to solve decoupled forward backward stochastic differential equations of McKean-Vlasov type, which appear in some stochastic control problems in a mean field environment. We give two algorithms and show that they have convergence of orders one and two under appropriate regularity conditions.
identifier: NNT: 2013NICE4115
issued: 2026-05-06T04:20:24.477501
language: en
modified: 2026-05-06T04:20:24.477506
contact point: Organization
  fn: CCSD
keyword: algorithm
keyword: algorithme
keyword: analyse-stochastique
keyword: champ-moyen
keyword: cubature
keyword: degeneracy
keyword: degenerescence
keyword: edp
keyword: eds
keyword: edspr
keyword: fbsde
keyword: infoeu-reposemanticsdoctoralthesis
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: mckean-vlasov
keyword: mean-field
keyword: numerical-scheme
keyword: parametrix
keyword: pde
keyword: resolubilite-forte
keyword: schema-numerique
keyword: sde
keyword: stochastic-analysis
keyword: strong-well-posedness
keyword: theses
landing page: https://theses.hal.science/tel-00954418

Stochastic differential equations : strong well-posedness of singular and degenerate equations; numerical analysis of decoupled forward backward systems of McKean-Vlasov type
type: Distribution
format: HTML
issued: 2026-05-06T04:20:24.509871
modified: 2026-05-06T04:20:24.437224
access URL: https://theses.hal.science/tel-00954418

test_moissonnage_selune
type: Agent