Is it possible to define a stochastic differential equation of the form
$$ dX(t) = f(t,X)dJ(t) $$
Where $J(t)$ is a compound Cox process defined like
$$ J(t) = \sum_{j=1}^{N(t,\Lambda)} \eta_j $$
With
- $N(t,\Lambda)$ - Poisson process with stochastic intensity $\Lambda(t)$, i.e. a Cox process;
- $\Lambda(t)$ - another stochastic process, that apparently (from reading a few articles) needs to have $\Lambda(0)=0$, $\Lambda(t)<\infty$ a.s., non-decreasing and right-continuous;
- $\eta$ - i.i.d. random variables, independent from everything else, that have values on some set $A$ (just to try and be more general and assume point processes).
If I'm not mistaken, this would imply the existence of a stochastic integral of the type
$$ \int_t^T f(s,X_{-})dJ(s)=\int_{(t,T]\times A} f(s,X_{-}) \mu_{J}(ds, d\eta)=f(t,X_{-})\sum_{s\in(t,T]} \Delta J(s) $$
Where the compound Cox jump sizes $\Delta J(s)$ are as implied by its expression, previously written.
Does an integral of this type exist for this type of processes, and can we numerically simulate them, as if they were more common compound Poisson process? Or is there some property that limits what we can do with them because they explode/diverge somewhere?
Furthermore, is it possible to define a compensated compound Cox process $\tilde{J}(t) = J(t) - \mathbb{E}[J(t)]$ that is a martingale, similar to what we can do with compound Poisson processes?
In this case, if I additionaly assume, for simplicity and illustration, that we know the distribution of the increments of $\Lambda(t)$ with some $\Lambda(t) - \Lambda(s) \sim U_{\Lambda}(t-s)$, we would have for the expectation of $J(t)$ that
$$ \mathbb{E}\left[J(t)\right] = \mathbb{E}\left[ N(t, \Lambda)\right] \mathbb{E}\left[\eta\right] = \mathbb{E}\left[\eta\right] \int_0^{\infty} k \frac{\lambda^k t^k e^{-\lambda} }{k!}\mu_{\Lambda}(d\lambda) $$
Where we know that $\mu_{\Lambda}$ will depend on $t$ because of the previous assumption. This looks like we will have a sort of convolution of the chosen $\mu_{\Lambda}$ by the gamma function. Would computing this then assure us that $\tilde{J}(t)$ would be a martingale?
EDIT: I tried to add a few more details from my line of reasoning and tried to expand on my thoughts on calculating the compensated version of this process.