Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$ is a constant. (default intensity)

Let $w_t^{i, \mathbb{P}} := H_t^i - \int_0^t (1-H_u^i) h_i^{\mathbb{P}}du$ a jump martingale.

Then the dynamics under the new measure $\mathbb{Q}$ is given by $dP_t^i = r_D P_t^i dt - P_{t-}^i dw_t^{i,Q}$ with $h_i^{\mathbb{Q}} = r^i - r_D + h_i^{\mathbb{P}}$

The Radon-Nikodym density is given as follows: $\frac{d\mathbb{Q}}{d\mathbb{P}}\Bigg \vert_{\mathcal{G_\tau}} = e^{\frac{r_D-\mu}{\sigma}W_\tau^{\mathbb{P}}-\frac{(r_D-\mu)^2}{2\sigma^2}\tau} \bigg(1 + \frac{r^I - r_D}{h_I^{\mathbb{P}}}\bigg)^{H_\tau^I} e^{(r_D-r^I) \tau} \bigg(1+ \frac{r^C - r_D}{h_C^{\mathbb{P}}}\bigg)^{H_\tau^C} e^{(r_D-r^C) \tau}$

They say that they apply Ito in the first formula for $dP_t^i$ and get the new dynamics with the new measure.

I don't get the new dynamics. How to apply Ito's formula for jump diffusions?


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