I have been reading chapter 2 on semimartingales in Jacod & Shiryaev. In order to motivate the semimartingale characteristics, they first take a look at the Lévy exponent for a Lévy process $X$ given as $$ \Psi_{t}(u) = iub_{t}-\frac{u^2}{2}c_t+\int (e^{iux}-1-iuh(x) )F_{t}(dx) $$ From stationarity of the increments of $X$, it then follows that $$ \frac{\exp(iuX_{t})}{\exp (\Psi_{t}(u))} $$ is a martingale. They then go on to say that it is possible to expand this notion of a Lévy characteristic to the semimartingale characteristic, such that it still satisfies this martingale property. I am also interested in why this property is even interesting or important to begin with. Why do we wish to preserve this?
However, I fail to see when they touch upon this again, or why this should somehow be obvious (if it is).
In their theorem 2.42, they prove that $X$ is a semimartingale with characteristics $(B,C,\nu)$ if and only if the following process is a local martingale \begin{align}\tag{1} e^{iuX_{t}}-\int_{0}^{t}e^{iuX_{s}}\text{d}A(u)_{s} \end{align} with (keeping things univariate) $$ A(u)_{t} = iuB_{t}-\frac{u^2}{2}C_{t} + \int(e^{iux}-1-iuh(x))\nu ([0,t]\times dx) $$ In the book it is stated that this should make the first statements of the chapter more precise, which I interpret to be the ones about the ratio $\exp (iuX_{t})/\exp (\Psi_{t}(u))$ being a martingale. Are these two things somehow equivalent? Or am I misunderstanding something?
EDIT: I just realised that the expression in (1) reminds me of the infinitesimal generator $\mathcal{A}$ for a Markov process $X$, which has the property that $$ f(X_{t}) - \int_{0}^{t}\mathcal{A}f(X_{s})ds $$ is a martingale. So for $f(x) = e^{iux}$ we get something similar to (1). For a Lévy process the infinitesimal generator is known in an analytic form and according to Wikipedia it satisfies $$ \mathcal{A}f(x) = -\int e^{iux}\psi (u) \hat{f}(u)d u $$ when $\psi (t)$ is the characteristic triplet of the process at time $t$. Perhaps some of this could be useful.
So it turns out that the postulated generalization is actually proved later on in the book. The proof is rather long, but essentially we have, that if we define the stochastic exponential of a semimartingale $X$ as $$ \mathcal{E}(X)_{t} = e^{X_{t}-X_{0}-\frac{1}{2}\langle X^{c},X^{c}\rangle_{t}}\prod_{s\leq t}(1+\Delta X_{s})e^{-\Delta X_{s}} $$ where $\langle X^{c},X^{c}\rangle_{t}$ is the quadratic variation of the continuous martingale part of $X$. This exponential is the solution to the stochastic differential equation $$ dY = YdX, \quad Y_0 = 1 $$ Then they prove under some assumptions on the jumps, that $X$ is a semimartingale with characterics $(B,C,\nu)$ if and only if the process $$ \frac{e^{iuX_{t}}}{\mathcal{E}(A(u))_{t}} $$ with $$ A(u)_{t} = iuB_{t} - \frac{u^2}{2}C_{t} + \int (e^{iux}-1-iuh(x))\nu ([0,t]\times dx) $$ is a local martingale. This generalizes the result I asked about in the sense that the ordinary exponential is simply exchanged with the stochastic exponential and some slight regularity on the jumps of $X$ (and we then obtain only a \emph{local} martingale).