There are different characterisations of a Markov Process: Probability Semigroups, Generators, even in some cases by Jumps Chains and Holding Times... And I know that, in "real life", the only thing one has is a Stochastic Differential Equation with a Markovian Behaviour...
My questions are:
1) Is the latter the most frequent or which one is the most common when dealing with real-models?
2) which is the relationship between the "SDE characterisation" and the rest? ( or where I can find information about it?). I'm trying know if there is any general presentation of a Markov Process as a solution of an SDE, and in which cases there is a correspondence between this characterisation and any of the other.
I don't know if it helps, but I'm dealing with a process like this:
\begin{equation} \begin{split} dX_i(t) = - X_i(t)dt & + \sum_{j \neq i} \int_{R_+^2} z_i \mathbb{1}_{\{ 0 \leq u \leq b(X_j(t-))\}} \mathcal{N}_j (du,dz,dt) \\ & - X_i(t^-)\int_{R_+^2} \mathbb{1}_{\{ 0 \leq u \leq b(X_i(t-))\}} \mathcal{N}_i (du,dz,dt) \end{split} \end{equation} Where $(\mathcal{N}_j)$ are independent Poisson Processes with intensity measure $du\bigotimes \mathbb{W}^j(dz)\bigotimes dt $, with \begin{equation} \mathbb{W}^j (dz) = \bigotimes _{k = 1} ^{j-1} W(dz_k) \bigotimes \delta_0(dz_j)\bigotimes_{k=j+1}^N W(dz_k) \end{equation}
$W(dx)$ non-negative, integrable and $b:R_+ \to R_+$ is $C^1$, non-decreasing.