Abstracts

We study a class of BSDEs, key to the modeling of counterparty risk in finance, with terminal time $ T \wedge S,$ where $T$ is a positive constant and $S$ is a stopping time. When $S$ has an intensity and under a decomposability assumption, so-called condition (B), on the model filtration $\mathbb{G}$, we prove that this BSDE is equivalent to a simpler BSDE, relative to a reduced filtration $\mathbb{F}$ and with constant terminal time $T.$ To prove this result, we generalize several results in Jeulin's classical theory of enlargement of filtrations, using new arguments (the classical ones can no longer be used at this increased level of generality). In addition, assuming the existence of a changed probability measure $\mathbb{P}$ equivalent to the original probability measure $\mathbb{F}$ on $\mathcal{F}_T$ such that any $({\mathbb{F}},\mathbb{P})$-local martingale {stopped at ${S-}$} is a $(\mathbb{G},\mathbb{Q})$-local martingale, the so-called condition (A), we prove that the previous $({\mathbb{F}},\mathbb{Q})$ reduced BSDE is in turn equivalent to an $(\mathbb{F},\mathbb{P})$ BSDE that is essentially the original $(\mathbb{G},\mathbb{Q})$ BSDE, but with the constant terminal time $T$ --- an equivalence that we call the invariance principle. In order to interpret the condition (A) in a classical langage of enlargement of filtration, we establish a characterization of this condition in terms of the Azéma supermartingale of $S.$ Thus we can compare the setup offered by conditions (A) and (B) with other models, such as pseudo stopping times or density models. Finally, we show that the case of a predictable time $S$, important for application to the so-called cure period (which relates to gap risk) in counterparty risk modeling, is of a completely different nature from the above case where $S$ had an intensity --- an important message for financial practitioners. In order to deal with the predictable case, we propose a technique of desintegration of the BSDE with respect to an intermediate stopping time. In the end, as the last section illustrates, conditions (A) and (B) appear as the right framework for the study of the counterparty risk related BSDEs.