A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE

Antoine Lejay 1, 2
1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We extend some results on time-homogeneous processes generated by divergence form operators to time-inhomogeneous ones. These results concern the decomposition of such processes as Dirichlet process, with an explicit expression for the term of zero-quadratic variation. Moreover, we extend some results on the Itô formula and BSDEs related to weak solutions of PDEs, and we study the case of quasi-linear PDEs. Finally, our results are used to prove the existence of weak solutions to forward–backward stochastic differential equations.
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Antoine Lejay. A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE. Stochastic Processes and their Applications, Elsevier, 2004, 110 (1), pp.145-176. ⟨10.1016/j.spa.2003.09.012⟩. ⟨inria-00001228⟩

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