Existence, Uniqueness and Regularity of Decoupling Fields to Multidimensional Fully Coupled FBSDEs

We develop an existence, uniqueness and regularity theory for general multidimensional strongly coupled FBSDE using so called decoupling fields. We begin with a local result and extend it to a global theory via concatenation. The cornerstone of the global theory is the so called maximal interval which is, roughly speaking, the largest interval on which reasonable solutions exist. A method to verify that the maximal interval is the whole interval, for problems in which this is conjectured, is proposed. As part of our study of the regularity of solutions constructed we show variational differentiability under Lipschitz assumptions. Extra emphasis is put on the more special Markovian case in which assumptions on the Lipschitz continuity for the FBSDE can be weakened to local ones, and additional regularity properties emerge.