Roughly speaking, when we have the dynamics of a stock process be given by the SDE: $$ dS_t = \mu_tS_tdt + \sigma_tS_tdW_t,$$ and if we can choose a cash account $B_t = e^{\int_0^t r_s ds}$ as numéraire and calculate the discounted stock process SDE $$d(\frac{S_t}{B_t})=\sigma_t \frac{S_t}{B_t}\bigg[\bigg(\frac{\mu_t - r_t}{\sigma_t} \bigg)dt + dW_t \bigg],$$ then by letting $dW^\mathbb{Q}_t=\frac{\mu_t - r_t}{\sigma_t} dt + dW_t$, the discounted stock process is a martingale under the equivalent measure $\mathbb{Q}$ and the Radon–Nikodym density can be calculated using the Doléans-Dade exponential as $$\frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{F}_T} = \mathcal{E}\bigg(\int_0^T \frac{\mu_s - r_s}{\sigma_s} dW_s\bigg)=\exp\bigg(\int_0^T \frac{\mu_s - r_s}{\sigma_s}dW_s - \frac{1}{2}\int_0^T\big(\frac{\mu_s - r_s}{\sigma_s}\big)^2 ds \bigg)$$ and the stock dynamics under $\mathbb{Q}$ are given by $$dS_t = r_tS_tdt + \sigma_tS_tdW_t^{\mathbb{Q}}.$$
My question is how can I apply the same steps for the given defaultable bond? $$dP_t = \bar{\mu}_t P_tdt-P_{t-}d \ \mathbb{I}_{\{\tau\le t\}}$$ assuming it holds that $$M_t = \mathbb{I}_{\{\tau \le t\}}- \int_0^t(1- \mathbb{I}_{\{\tau \le s\}})\lambda_s \ ds \quad \mbox{is a ($\mathbb{G},\mathbb{P}$)-martingale}$$ where $\mathbb{I}_{\{\tau\le t\}}$ is an indicator function (my main difficulty) and $\tau$ is a random time in $\mathbb{F}$ (independent exponentially distributed random variable that is not an $\mathbb{F}$-stopping time and has intensity $\lambda_t$ ) and $\mathbb{G}$ is a progressively enlarged filtration such that $\mathbb{F} \subset \mathbb{G}$ and $\tau$ is a $\mathbb{G}$-stopping time, and $\bar{\mu}$ is deterministic.
To be more specific, how can I calculate the following $$d(\frac{P_t}{B_t})=?, \quad M^{\mathbb{Q}}_t =?, \quad \frac{d\mathbb{Q}}{d\mathbb{P}}\bigg|_{\mathcal{G}_\tau}= \mathcal{E}\bigg(\int_0^\tau \ (?) d \ \mathbb{I}_{\{\tau\le s\}}\bigg)=?$$