Differentiability of Quadratic BSDEs

We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. This Parameter $x$ can be viewed as the initial condition of an SDE, opening the way to differentiability of forward-backward systems. Finally we 'prove' sufficient conditions for solutions of quadratic BSDE to be differentiable in the variational sense (Malliavin differentiable) and give representation formulas. These proofs are very technical, so we will show simple sketches of them focusing on the tools we used.