# Risk measures and progressive enlargement of filtration: a BSDE approach

## TL;DR

This paper develops a framework for dynamic risk measures using BSDEs in an enlarged filtration setting, incorporating default times and marks, with applications to financial risk management and explicit examples.

## Contribution

It introduces a novel BSDE-based approach to dynamic risk measures in enlarged filtrations, decomposing risk before and after default times with explicit examples.

## Key findings

- Decomposition of risk measures into pre- and post-default components.
- Characterization of risk measure properties under BSDE assumptions.
- Explicit examples and numerical simulations of entropic risk measures.

## Abstract

We consider dynamic risk measures induced by Backward Stochastic Differential Equations (BSDEs) in enlargement of filtration setting. On a fixed probability space, we are given a standard Brownian motion and a pair of random variables $(\tau, \zeta) \in (0,+\infty) \times E$, 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 $(\tau, \zeta)$, we define the dynamic risk measure $(\rho_t)_{t \in [0,T]}$, for a fixed time $T > 0$, induced by its solution. We prove that $(\rho_t)_{t \in [0,T]}$ can be decomposed in a pair of risk measures, acting before and after $\tau$ 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 $(\rho_t)_{t \in [0,T]}$ and we discuss the dynamic entropic risk measure case, providing examples where it is possible to write explicitly its decomposition and simulate it numerically.