The dynamic spread of the forward CDS with general random loss (Q1724436)

Summary: We assume that the filtration \(\mathbb{F}\) is generated by a \(d\)-dimensional Brownian motion \(W = (W_1, \ldots, W_d)'\) as well as an integer-valued random measure \(\mu(d u, d y)\). The random variable \(\widetilde{\tau}\) is the default time and \(L\) is the default loss. Let \(\mathbb{G} = \{\mathcal{G}_t; t \geq 0 \}\) be the progressive enlargement of \(\mathbb{F}\) by \((\widetilde{\tau}, L)\); that is, \(\mathbb{G}\) is the smallest filtration including \(\mathbb{F}\) such that \(\widetilde{\tau}\) is a \(\mathbb{G}\)-stopping time and \(L\) is \(\mathcal{G}_{\widetilde{\tau}}\)-measurable. We mainly consider the forward CDS with loss in the framework of stochastic interest rates whose term structures are modeled by the Heath-Jarrow-Morton approach with jumps under the general conditional density hypothesis. We describe the dynamics of the defaultable bond in \(\mathbb{G}\) and the forward CDS with random loss explicitly by the BSDEs method.