Gussian diffusions and continuous state branching processes with killing

Abstract

This work considers a diffusion process , subject to killing at a rate which depends on time and current value of a diffusion. A conditioned process is defined, on , by restricting the sample space of , to the sample paths which survived up to time t. It is shown that conditioned process is a diffusion whose infinitesimal parameters are functions of the infinitesimal parameters of , and of the survival function of . This result is applied to the problem of identification of a parametric form of a killing-rate function such that an unconditioned and an associated conditioned diffusion belong to the same family of diffusions. The specific results are obtained for Gaussian diffusions and for continuous state branching processes. For these diffusions the required killing-rate functions are identified and the properties of associated diffusions are discussed. In particular formulas for marginal survival functions of these diffusions are obtained. A general formula for the survival function of a diffusion when the values of a diffusion are known at discrete times is also discussed. The applications of this work to survival analysis and possible extensions are indicated