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Backward stochastic differential equations driven by Gaussian Volterra processes

Abstract : This thesis treats of backward stochastic differential equations (BSDE) driven by a class of Gaussian Volterra processes that includes multifractional Brownian motion and multifractional Ornstein-Uhlenbeck processes. In the first part we study multidimensional BSDE with generators that are linear functions of the solution. By means of an Itoˆ formula for Volterra processes, a linear second order partial differential equation (PDE) with terminal condition is associated to the BSDE. Under an integrability condition on a functional of the second moment of the Volterra process in a neighbourhood of the terminal time, we solve the associated PDE explicitely and deduce the solution of the linear BSDE. We discuss an application in the context of self-financing trading stategies. The second part of the thesis treats of non-linear BSDE driven by the same class of Gaussian Volterra processes. The main results are the existence and uniqueness of the solution in a space of regular functionals of the Volterra process, and a comparison theorem for the solutions of BSDE. We give two proofs for the existence and uniqueness of the solution, one is based on the associated PDE and a second one without making reference to this PDE, but with probabilistic and functional theoretic methods. Especially this second proof is technically quite complex, and, due to the absence of mar- tingale properties in the context of Volterra processes, requires to work with different norms on the underlying Hilbert space that is defined by the kernel of the Volterra process. For the construction of the solution we need the notion of quasi-conditional expectation, a Clark-Ocone type formula and another Itoˆ formula for Volterra processes. Contrary to the more classical cases of BSDE driven by Brownian or fractional Brownian motion, an assumption on the behaviour of the kernel of the driv- ing Volterra process is in general necessary for the wellposedness of the BSDE. For multifractional Brownian motion this assumption is closely related to the behaviour of the Hurst function.
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Submitted on : Wednesday, June 10, 2020 - 2:28:32 PM
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H Knani. Backward stochastic differential equations driven by Gaussian Volterra processes. Mathematics [math]. Université de Lorraine; Université du Centre (Sousse, Tunisie), 2020. English. ⟨NNT : 2020LORR0014⟩. ⟨tel-02863666⟩

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