Extended Poisson equation for weakly ergodic Markov processes (Q2849272)

The first part of the paper discusses Poisson equations \(Au=-f\), where the linear operator \(A\) is the extended generator of a general Markov process. Solvability conditions for such equations are derived. In the second part of the present paper, the authors study the structure of the predictable component in the Doob-Meyer decomposition of processes of the form \(g(X(t),Y(t))\), \(t\geq 0\). Here \(X\) is a general Markov process and \(Y\) is the solution of a stochastic differential equation (SDE) driven by a Wiener process and a compensated Poisson random measure. The process \(Y\) depends on \(X\) as the coefficients of the SDE defining \(Y\) depend on \(X\). The function \(g\) is a function of the form as those arising as solutions to the Poisson equation studied in the first part of the paper.