Risk Measures and Progressive Enlargement of Filtration: A BSDE Approach

We consider dynamic risk measures induced by backward stochastic differential equations (BSDEs) in an enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $( au, zeta) in (0,+infty) imes E$, with $E subset mathbb{R}^m$, that enlarge the reference filtration, i.e., the one generated by the Brownian motion. These random variables can be interpreted financially as a default time and an associated mark. After introducing a BSDE driven by the Brownian motion and the random measure associated to $( au, zeta)$, we define the dynamic risk measure $( ho_t)_{t in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $( ho_t)_{t in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $ au$, and we characterize its properties giving suitable assumptions on the driver of the BSDE. Furthermore, we prove an inequality satisfied by the penalty term associated to the robust representation of $( ho_t)_{t in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.