Let $X(t)$ be a pure jump process. If we have a function $f(X_t)$ then Itô's formula for pure jump processes tells us that $$df(X_t) = \int_{\mathbb{R}}\left[f(X_{t_{-}} + x) - f(X_{t_{-}})\right]N_X(dt,dx)$$
How can we calculate the characteristic function for such a resulting process when we cannot make the reduction $$f(X_{t_{-}} + x) - f(X_{t_{-}}) = g(x) ?$$ In other words, in the case that we need to know the value of $X_{t_{-}}$ to calculate the size of the jumps.
I believe an example could be $f(X_{t}) = \left(a+bX_{t}\right)^p$, for a real power $p$.
If I understand it correctly, from Campbell's formula, when we can make that reduction above, we have \begin{aligned} \begin{split} \mathbb{E}\left[ \exp\left(iu \int_0^t \int_{\mathbb{R}}g(x)N_X(ds,dx)\right)\right] &= \exp \left( \int_0^t \int_{\mathbb{R}} \left( \mathrm{e}^{iug(x)} - 1 \right) \nu_X(dx)ds\right) \\ &= \exp \left( t \int_{\mathbb{R}} \left( \mathrm{e}^{iug(x)} - 1 \right) \nu_X(dx)\right) \end{split} \end{aligned} Where $\nu_X(dx)$ is the intensity measure of $X_t$.
I'm not sure if this implies that the characteristic function conditional on $X_s$ can be written
$$\mathbb{E}\left[ \exp\left(iu \int_0^t \int_{\mathbb{R}}\left[f(X_{s_{-}} + x) - f(X_{s_{-}})\right]N_X(ds,dx)\right)\bigg \vert X_{s_{-}}\right]=\exp \left( \int_0^t \int_{\mathbb{R}} \left( \mathrm{e}^{iu\left[f(X_{s_{-}} + x) - f(X_{s_{-}})\right]} - 1 \right) \nu_X(dx)ds\right)$$
But even if this is the case, I don't know how to further proceed from here; not only because I don't see how to include the information $X_s = \int_0^s \int_{\mathbb{R}} xN_X(dr,dx)$ but also because I don't see how to write the analogous expression for $X_{s_{-}}$ (how can we remove just the effect of the very last jump?).
Even if we rewrite it to be conditional on $X_s$ instead
$$\mathbb{E}\left[ \exp\left(iu \int_0^t \int_{\mathbb{R}}\left[f(X_s) - f(X_s - x)\right]N_X(ds,dx)\right)\bigg \vert X_s\right]=\exp \left( \int_0^t \int_{\mathbb{R}} \left( \mathrm{e}^{iu\left[f(X_s) - f(X_s - x)\right]} - 1 \right) \nu_X(dx)ds\right)$$
The problems remain, and in this case I think $x$ cannot be independent of $X_s$ because it's the size of the jump that already occurred in $s$.