Assume that \(\tau\) is an exponential r.v. with parameter \(\lambda\) and that \(\mathbf{G}\) is the filtration generated by \(\mathbb{1}_{\{\tau \leq t\}}\). Check that \(\lambda_{t}^{\mathbf{G}}=\mathbb{1}_{\{t<\tau\}} \lambda\). Let \(Y_{t}=\mathbb{E}\left(\exp -(T \wedge \tau) \mid \mathcal{G}_{t}\right)\). Compute the jump of \(Y\) at time \(\tau\) and deduce \(\mathbb{E}\left(\mathbb{1}_{\{t<\tau \leq T\}} \mid \mathcal{G}_{t}\right)\) using Proposition 7.7.2.1 . Of course, here, the conditional survival methodology is more powerful.

Proposition 7.7.2.1:

For every integrable r.v. \(X \in \mathcal{G}_{T}\) :

\[\mathbb{E}\left(X \mathbb{1}_{\{T<\tau\}} \mid \mathcal{G}_{t}\right)=\mathbb{1}_{\{\tau>t\}}\left(V_{t}-\mathbb{E}\left(\Delta V_{\tau} \mathbb{1}_{\{\tau \leq T\}} \mid \mathcal{G}_{t}\right)\right)\]

where \(V_{t}=e^{\Lambda_{t}^{\mathrm{G}}} \mathbb{E}\left(X e^{-\Lambda_{T}^{\mathrm{G}}} \mid \mathcal{G}_{t}\right)\).