Abelian-type expansions and non-linear death processes(II)

This paper is concerned with the study of death processes with time-homogeneous non-linear death rates. An explicit formula is obtained for the joint distribution of the state X T and the variable ∫ T 0 g ( X t ), where g is any given real function and T corresponds to some appropriate stopping time. This is achieved by constructing a family of martingales and then by using a particular family of Abel–Gontcharoff pseudopolynomials (the theory of which has been introduced in a companion paper) and related Abelian-type expansions. Moreover, the distribution of the first crossing level of such a death process through a general upper boundary is also evaluated in terms of pseudopolynomials of that kind. The flexibility of the methods developed makes easy the extension to multidimensional death processes.