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 some constant.
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 not relevant here. 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. Can somebody help?