On~the existence of a solution in a problem of controlling a~counting process

Yu. M. Kabanov

Sbornik. Mathematics, Tome 47 (1984) no. 2, pp. 425-438

Résumé

An existence theorem is proved in the control problem $\mathbf E^u\xi\to\max$, where $\xi$ is a bounded functional of the sample functions of a counting process $x=(x_t)_{t\geqslant0}$ with intensity $\lambda^u=\lambda(x,t,u(x,t))$. It is assumed that $\xi$ satisfies a certain condition of weak dependence on the “tail” of the sample function. The proof is based on compactness considerations and makes essential use of a description of the extreme points of the set of admissible local densities. The Appendix gives a description of the set of extreme points for the family of distribution densities of diffusion-type processes relative to Wiener measure. Bibliography: 17 titles.