Asymptotic behavior of Markov evolutions until absorption
Let \(X_{\epsilon}(t)\) be a homogeneous Markov process with absorption state 0, defined by the Markov kernel \[ Q_{\epsilon}(x,A,t)=P_{\epsilon}(x,A)(1-e^{-\alpha (x)t}),\quad P_{\epsilon}(x,A)=P_ 0(x,A)-\epsilon P_ 1(x,A). \] Let \(y_{\epsilon}(t,x)\) be a family of homogeneous continuous processes with independent increments and uniformly bounded cumulants \(\psi (x,z)=\ln M(e^{izy(t,x)})/t.\) This paper gives an algorithm for the construction of an asymptotic expansion for the characteristic function of the Markov random evolution \(y_{\epsilon}(t)\) formed by \(y_{\epsilon}(t,x)\) and switched by \(X_{\epsilon}(t):\) \[ U_{\epsilon}(t)=e^{-\Lambda t}+\sum_{k\geq 1}\epsilon^ k[e^{-\Lambda t}U_ k(t)+V_ k(t/\epsilon)]. \] Where \(U_ k(t)\) and \(V_ k(t)\) are constructed recursively, \[ \Lambda =\int_{E}\pi (dx)[\alpha (x)P_ 1(x,E)-\psi (x,z)], \] \(\pi\) is a measure related to \(\alpha(x)\) and a stationary distribution of an unperturbed, uniformly ergodic Markov chain defined by \(P_ 0(x,\cdot)\).
asymptotic expansion for the characteristic function
scientific article; zbMATH DE number 3982204