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.
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