I am trying to understand an argument involving the pricing kernel $\xi_t$ in the context of a simple jump diffusion model for the price of an asset $S_t$: \begin{align} \xi_t = \exp \left[ -\theta W_t - \left( \frac{\theta^2}{2} + \lambda \kappa' \right)t + \sum_{n=1}^{N_t} (\gamma J_n + \nu) \right] \\ \frac{dS_t}{S_{t^-}} = (\mu -\lambda \kappa) dt + \sigma dW_t + (e^J - 1) dN_t \end{align} where $W_t, N_t, J$ are mutually independent and $N_t$ follows a Poisson process such that $E^P((e^J - 1)dN_t) = E^P(e^J - 1) E^P(dN_t) = \kappa \lambda dt$. We also have $\kappa' = e^\nu E^P(\exp(\gamma J)) - 1$.
In that model, under the risk-neutral measure $Q$, we should find among other things these new parameter values for the jump part: $\kappa^* = E^Q(e^{J^*} - 1) \text{ and } \lambda^* = \lambda( \kappa' + 1)$. Now, I am not too familiar with jump-diffusion models, but to exploit my definition of $\kappa'$ later on, I would need to find something like this: \begin{equation} \frac{d\xi_t}{\xi_{t^-}} = - \theta dW_t - \left( \frac{1}{2}\theta^2 + \lambda \kappa' \right)dt + \exp(\gamma J + \nu)dN_t \end{equation} because I plan on exploiting the Euler equation under $P$: \begin{equation} E^P \left( \frac{d(\xi_t A_t)}{\xi_{t^-} A_{t^-}} \right) = 0 \end{equation} where $A_t$ is the price of some asset. Here, I'll be using a riskless bond to find a relationship with the riskless rate of return and, obviously, the stock price whose dynamics under $P$ is given above. In principle, this gives me the mapping between the parameters of $\xi_t$ and those of the model for $S_t$ (and the riskless rate of return).
Is it correct and, if so, can someone explicit the steps required to go from the pricing kernel to the differential equation describing its dynamics? I suppose I could show that the level process solves the SDE, but I want to know how to move from the level process to the SDE.
I am able to do it, unfortunately, only if $\gamma, \nu = 0$.