Markov jump-process generator

For transition rates $q_{nm}$, the generator acts on a function $f$ byWhenever the expectations are finite,Choosing polynomial functions $f$ gives differential equations for the moments.

For a positive function $g$ in the domain of a Markov jump-process generator, the generator martingale for $g(X_t)$ and semimartingale integration by parts give the displayed local martingale. Suitable finite-horizon deterministic bounds on $g$ and $Lg/g$ make it a true martingale. For a single marked exponential jump process, take $g(x)=e^{\theta x}$. Then the exponent's compensating process is $r_{\theta }(t\wedge T)$, where $r_{\theta }=(e^{\theta }-1)/\theta -1$ for $\theta \ne 0$ and $r_0=0$; it is continuous adapted and hence predictable.

For jumps $x\mapsto x+{\nu }_r$ of rates $a_r(x)$, the forward operator acts on a probability mass function byIt is adjoint to the Markov jump-process generator $Lf(x)={\sum }_r{a}_r(x)[f(x+{\nu }_r)-f(x)]$.