Time-delayed BSDEs

Abstract:

We start with an excursus on Backward Stochastic Differential Equations (BSDEs) with a particular emphasis on BSDEs with time-delayed generators. We consider FBSDEs driven by Lévy noise, the central focus being the derivation of a non-linear Feynman–Kac representation formula constituting a bridge between the FBSDEs and the solutions of a path-dependent non-linear Kolmogorov equation. Then, we analyse BSDEs with a Stieltjes integral term and small delays, deriving the conditions under which the existence and uniqueness of solutions are guaranteed. We extend these results on arbitrary delays, deriving the necessary conditions, monotonicity and linearity, for the generators. In the last part, we explore a coupled system of FBSDEs consisting of a reflected SDE and a BSDE with a Stieltjes integral. This system is linked to a non-linear path-dependent PDE with homogeneous Neumann boundary conditions. Throughout the seminar, the theoretical advancements are contextualized within practical applications in finance and actuarial science.