Decoupled mild solutions of deterministic evolution problemswith singular or path-dependent coefficients, represented by backward SDEs

Abstract : This thesis introduces a new notion of solution for deterministic non-linear evolution equations, called decoupled mild solution.We revisit the links between Markovian Brownian Backward stochastic differential equations (BSDEs) and parabolic semilinear PDEs showing that under very mild assumptions, the BSDEs produce a unique decoupled mild solution of some PDE.We extend this result to many other deterministic equations such asPseudo-PDEs, Integro-PDEs, PDEs with distributional drift or path-dependent(I)PDEs. The solutions of those equations are represented throughBSDEs which may either be without driving martingale, or drivenby cadlag martingales. In particular this thesis solves the so calledidentification problem, which consists, in the case of classical Markovian Brownian BSDEs, to give an analytical meaning to the second component Z ofthe solution (Y,Z) of the BSDE. In the literature, Y generally determinesa so called viscosity solution and the identification problem is only solved when this viscosity solution has a minimal regularity.Our method allows to treat this problem even in the case of general (even non-Markovian) BSDEs with jumps.
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Adrien Barrasso. Decoupled mild solutions of deterministic evolution problemswith singular or path-dependent coefficients, represented by backward SDEs. Probability [math.PR]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLY009⟩. ⟨tel-01895487⟩

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